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MSU I. An Afﬁrmativ- Action/Equal Opponunhy Inﬂation
(Wm:

 

Essays on Growth and International Trade

By

Young-Min K won

A DISSERTATION

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Economics

1996

ABSTRACT

Essays on Growth and International Trade
By

Young-Min K won

Chapter 1 analyzes the growth effect of common tariffs using a two country ver-
sion of the model developed by Young [1993]. In one of the two equilibria in the
model, it is shown that an increase in the common tariff rate initially leads to faster
growth but eventually leads to slower growth as the tariff rate rises. As the tariff rate
approaches infinity, the growth rate asymptotically approaches the free trade growth
rate. The result in this complementarity dominant equilibrium, at which innovators
are pessimistic about the future, is the exact opposite of that obtained by Romer
and Rivera-Batiz [1991]. The key difference is that Romer and Rivera-Batiz assume
that newly developed products always take market share away from existing prod-
ucts. However, by allowing new products to complement old products, I show that
the complementarity dominant equilibrium behaves as explained above.

Chapter 2 presents a R&D-based growth model in which innovations are in terms
of intermediate product differentiation. Like in Jones [1995b] and Segerstrom [1995],
the model presented here is free of the scale effects found in many R&D-based growth
models. The long-run growth rate in this model is determined by the combination of
the endogenously determined human capital accumulation rate and the exogenously
given population growth rate. When consumption at two points in time is highly sub-
stitutable, an improvement in the educational environment lowers the human capital

growth rate and the long-run economic growth rate. A higher population growth rate

also induces slower economic growth for some median value of substitutability.
Finally in Chapter 3, using an endogenous growth model with North-South tech-
nology trade, I analyze the long-run growth effects of changes in labor supply. Inno-
vation occurs only in the North. Increases in human capital in each region have a
positive growth effect, whereas increases in northern unskilled labor negatively affect
the long-run growth rate. Effects of increases in southern unskilled labor could be
either positive or negative. Results in this chapter are new, compared to the imitation
model of Lai [1995], as changes in northern labor supply don’t have growth effects
in his paper. Results here are also different from Liu [1994], where only one type of

labor is considered and increases in labor always have a positive growth effect.

Copyright by
Young Min Kwon
April 29, 1996

To my wife and my children
For their Love, Support, and Sacriﬁce

ACKNOWLEDGMENTS

This dissertation is completed after valuable help from many people. First, I
would like to extend my deep thanks and gratitude to Professor Paul Segerstrom,
my advisor and committee chairperson. Without his time consuming guidance and
encouragement, this dissertation could never have been completed. I would extend
my appreciation to Professor Steven Matusz and Professor Carl Davidson for serving
as members of my dissertation committee.

The graduate student community within the department of economics has offered
valuable support, both intellectually and emotionally. Especially, I am indebted spe-
cial thanks to Jess Reaser and Hailong Qian, who share not only offices in the third
ﬂoor Marshall but also those moments of ups and downs during the long process of
my dissertation.

I also would like to express my special thanks to Ms. Ann Feldman. Without her
help in various stages during my stay here, the life would have been very difficult.

Finally, I give heartful thanks to my family. My parents, Seungchae Kwon and
Kyoungja Kim, always have been supportive and patient during the years of my
education. My parents-in-law, Deungho Choi and Soonim Hong, also have been very
warmful and supportive. Very many thanks to my wife, Seounghee Choi, and my

Children, Hoon and Jane, for their love, patient, and encouragement.

vi

Contents

List of Figures

1

viii

The Growth Effects of Tariffs When There is Inovation Complemen-

tarity

1

1.1 Introduction ................................ 1
1.2 The Model ................................. 4
1.2.1 Consumers ............................. 6
1.2.2 Final Goods Manufacturer .................... 8
1.2.3 Intermediate Goods ........................ 9
1.2.4 Research .............................. 14
1.2.5 Labor Market ........................... 15

1.3 Steady State Equilibria .......................... 16
1.4 Growth Effects of Tariffs ......................... 21
1.5 Comparison with Rivera-Batiz and Romer ............... 28
1.6 Conclusion ................................. 30
A R&D base Growth Model with Human Capital Accumulation 32
2.1 Introduction ................................ 32
2.2 The Model ................................. 34
2.2.1 Consumers ............................. 35
2.2.2 Manufacturing .......................... 37
2.2.3 Research .............................. 39

2.3 Steady State Equilibrium ......................... 40
2.4 Comparative Dynamics .......................... 49
2.5 Conclusion ................................. 58

Factor Endowments and Economic Growth in a Model With Tech-

nology ﬁade 61
3.1 Introduction ................................ 61
3.2 The Model ................................. 63
3.2.1 Consumers ............................. 64
3.2.2 Innovation ............................. 66
3.2.3 Northern Production ....................... 67
3.2.4 Southern Production ....................... 68

3.2.5 No Arbitrage Condition ..................... 7O

vii

viii

3.2.6 Labor Market ........................... 71

3.3 Steady State Equilibrium ......................... 72
3.4 Comparative Dynamics .......................... 79
3.4.1 Increases in Northern Human Capital (H N T) ......... 80

3.4.2 Increases in Northern Unskilled Labor (LN T) ......... 83

3.4.3 Increases in Southern Human Capital (H5 T) .......... 86

3.4.4 Increases in Southern Unskilled Labor (L3 T) .......... 87

3.5 Conclusion ................................. 90
Appendix 92
A Chapter 1 92
A.1 Consumers’ Expenditure Path ...................... 92
1°12 Final Goods Manufacturer ........................ 93

B Chapter 2 99
C Chapter 3 102
0.1 Slopes of the N N and the SS Curves .................. 103
C.2 Increases in Northern Human Capital (HN T) ............. 105
C.3 Increases in Northern Unskilled Labor (LN T) ............. 107
0.4 Increases in Southern Human Capital (H5 T) .............. 108
C.5 Increases in Southern Unskilled Labor (Ls T) .............. 109

List of Figures

1.1
1.2
1.3

2.1
2.2
2.3
2.4
2.5
2.6

3.1
3.2
3.3
3.4

U C and PR ................................ 18
Changes in the Growth Rate ....................... 22
Effects of tariff on U C and PR ...................... 23
Equilibrium Human Capital Growth Rate ............... 45
Effect of Increase in A when d > do ................... 50
Effect of Increase in A when d < do ................... 52
Effect of Increase in it when d > 0’] ................... 54
Effect of Increase in n when 1 < d < 0‘1 ................. 55
Effect of Increase in 11 when d < do ................... 56
Steady State Equilibrium ......................... 75
Growth Effect of H N T .......................... 81
Growth Effect of LN T .......................... 84
Growth Effect of H5 T .......................... 86

ix

Chapter 1

The Growth Effects of Tariffs
When There is Inovation

Complementarity

1 .1 Introduction

In trade between two countries, the conventional wisdom is that less obstacles to the
ﬂow of goods and services across the border is better for economic growth in both
countries. The ideas of free trade helping economic growth can be implicitly found as
early as Ricardo [1815]. Formalized rather recently by Samuelson [1959] and Pasinetti
[1960], Ricardo’s idea could be used for advocating free trade as it allocates resources
in the world economy more efficiently allowing countries to gain from the free ﬂow

of goods and services across borders. A detailed discussions of numerous theoretical

 

 

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efforts1 can be found in Findlay [1984]. Those theoretical backgrounds as well as the
past experiences like the Smooth-Hawley Act passed in the US. during the Great
Depression are well incorporated into the successive tariff reductions of the General
Agreement on Tariffs and Trade negotiations.

Assuming increasing return to scale and imperfect competitions, theoretical re-
search 2 in 19808, however, give a mixed view about free trade arguments. Even
though the new trade theory has not been able to give any conclusion, in Krugman’s
terma, it makes free trade irretrievably lose its innocence. That is, there now is a
possibility that free trade might not be the only universally desirable trade policy.
As recent models with endogenous technological progress also involve market imper-
fections and increasing returns to scale, new growth theory also gives inconclusive
results. By allowing for productivity differences between countries, Grossman and
Helpman[1990] show certain conditions under which a small increase in the tariff
rate increases the long-run growth rate. In their model, a small tariff increase by a
country causes opposite movements of resources in and out of the research sectors in
each country and the net effect depends on the productivity difference. Extending
the Romer[1990] model to accommodate international trade in a symmetric setting,
Rivera-Batiz and Romer[1991] show that the increase in the common tariff rate slows
the economic growth at lower tariff levels. As tariff rates keep on rising the trends of

slow growth reverse and eventually go back to the original free trade growth level. In

 

1For example, Uzawa [1961] and Vanek [1971] presented two-sector neoclassical growth models
that connect growth and trade.

”Such as Brander and Spencer [1985] and Krugman [1986].

3Krugman [1987].

 

 

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their model, new products only substitute old products in the sense that newly devel-
oped products always take market share away from previously introduced products.

New products, however, do not always take market share away from existing prod-
ucts. New inventions often complement the use of existing products. For example,
the introduction of a new advanced microprocessor used in personal computer sys-
tems allows consumers to have faster computers at more affordable prices. This can
boost not only the sales of computers themselves but also the usages of many other
existing parts used in the production of the computers. Often ignored in Schum-
peterian growth models, as Young [1993] pointed out, this rent creating feature of
newly introduced products may play an important role in determining the economy’s
growth rate. Thus, it is interesting to use Young’s model that allows for innovation
complementarity and to ask the same question as Rivera-Batiz and Romer did.

By modifying Young’s model for a two country setting, this paper shows that an
increase in the tariff rate from initially low levels can enhance the economic growth
rate for both countries at one of two possible equilibria. In fact, in this complemen-
tarity dominant equilibrium, increases in the tariff rate initially increase the growth
rate when the tariff rate is low, after which any further increase in the tariff rate
decreases the growth rate, which eventually approaches the free trade growth rate.
This result is exactly opposite to the Rivera-Batiz and Romer model, while the other
equilibrium with substitution dominance behaves as in their model.

The introduction of innovation complementarity to the standard Schumpeterian
growth model gives expectations about future innovation a crucial role in determining

the effect of tariffs on the long-run growth rate. When the economy is in the comple-

Ba

sag

4

mentarity dominant equilibrium where people are pessimistic about the future rate of
innovation, a reduction in the tariff rate at lower levels has a negative growth effect.
However, if the economy is in the substitution dominant equilibrium where people are
optimistic about the future rate of innovation, the reduction in the tariff rate at lower
levels will have a positive growth effect, as conventional free trade theories suggest.
Although the selection of a single equilibrium is a matter of how to incorporate the
information structure in a model and this in turn is an empirical matter, the intro-
duction of innovation complementarity enlarges the set of possibilities in considering
the effect of the change in the tariff rate on the economy’s long-run growth rate.

In Section 2 below, the structure of the model is described. The solution of the
model for steady state equilibria with positive growth rates is developed in Section
3. Then in Section 4, the effect of tariffs on the economic growth rate is analyzed.
Section 5 compares the differences in the intuitions between this model and the Rivera-

Batiz and Romer model. Finally, in Section 6, policy implications of the findings and

suggestions for future research are discussed.

1.2 The Model

The model here considers a ”home” country and a ”foreign” country whose economies
are symmetric in every respect. In each country, there are three types of economic
activity on the production side. Researchers develop new types of intermediate goods
that are used in the production of ﬁnal goods. If successful, they are awarded with

inﬁnite patent protection; The manufacturer of the intermediate good, who acquires

 

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the patent right from the researcher,‘ can charge the monopoly price for the product
to ﬁnal goods manufacturers. As a by-product of the development of the new inter-
mediate goods, it becomes known how to produce a new type of a ﬁnal good which
is manufactured by using all existing intermediate products.5 Since this knowledge
is known to everybody, perfect competition prevails in the ﬁnal goods manufacturing
sector.

With this speciﬁcation, the development of new intermediate goods has two effects.
The substitution effect of new product development occurs as market share is taken
away from the producers of existing intermediate goods in the sense that a wider
variety of intermediate goods now competes to share the existing level of expenditure.
However, as a new type of ﬁnal good, which uses all existing intermediate goods in its
production, emerges as a by-product, we have a new market for every intermediate
good. This is the complementarity effect.

A common ad valorem tariff of rate 6 is imposed on intermediate products that
cross the border.6 Tariff revenue collected by each importing country’s government
is redistributed back to its own consumers in a lump-sum fashion. Because of the
symmetry between the countries, we will have symmetric calculations for each country

on many occasions. In the analysis below, we will focus on the home country and for

 

‘Even though the sales of the patent right across the border is not prohibited, the symmetric
assumption of this model requires an equal variety of intermediate goods produced in each country.
For simplicity, however, we consider the model in which international sales of patent right does not
occur in the analysis below.

“In Young’s model, there is an upper bound for each intermediate good can be used in ﬁnal
goods production. That is, a production of a ﬁnal good there uses only relatively new intermediate
products. For simplicity, we remove the limit in our model.

“We assume that there exists transportation costs of ﬁnal goods across the border. With perfect

competitions in the ﬁnal goods sector, the presence of transportation costs then prohibit the trade
of final goods between countries.

 

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the most part omit the corresponding calculations for the foreign country.

1.2.1 Consumers

Consumers in each country live forever and try to maximize their intertemporal utility

U E [‘00 e"”("‘)v(s)ds (1.1)

where the consumer’s subjective discount rate, p, is greater than 0 and the instanta-

neous utility v(t) is given by

nu) , , "(0 .. ..
time/o In0(1,t)d]+jo lnC’(J,t)dJ (1.2)

where C' (j, t) is the consumption a ﬁnal good, j, originated" from the home country
and C ( j“, t) is the consumption of a ﬁnal good, j", originated from the foreign country.
Note that, at time t, n(t) products are originated from each country.

As the intertemporal utility function is time separable, the consumer maximiza-
tion problem can be solved in two stages. That is, we can ﬁrst maximize for the
instantaneous utility with a given expenditure level and then solve for a equilibrium
time path for consumers’ expenditure. Since v(t) is logarithmic utility, the consumers
will evenly distribute their expenditure, E (t), on each product. Thus denoting j to

be a index of a home originated ﬁnal product, the expenditure for the ﬁnal good,

E(j,t), is given as

 

7Final goods produced in a county are originated either from the home or the foreign counties as
intermediate goods are developed.

 

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As shown in Appendix, using (1.3) into (1.2), (1.2) in (1.1) and then maximizing

(1.1) with respect to an intertemporal budget constraint, we obtain that the con-

sumers’ optimal expenditure path must satisfy

$7) = Re) — p + 90) (1.4)

where R(t) is the cumulative interest rate and g(t) E n’(t)/n(t) is the growth rate of
product variety at time t.

Since the preference represented by (1.1) and (1.2) is homothetic, the aggregate
expenditure is proportional to that of a representative consumer. Therefore, if we
consider E without subscripts t as the total expenditure level for a country,“ a path
satisfying (1.4) can also describe the whole economy.

If we normalize this economy-wide expenditure as E = E' = 1, we will have

R’=p-g (1.5)

Note that the presence of logarithmic signs in front of C (-)s in (1.2) make the failure of
consuming any variety lead to inﬁnitely negative utilityg. Consumers in this economy

have a tendency to postpone their expenditure to the future as the growth rate, 9,

 

3We will drop the time subscript t in the analysis below.

9As Young himself pointed out, these preferences exhibit exceedingly strong preferences for vari-
ety. However, we can show that basic features of the model presented here carried over to the more
complicated model with more general form of a CBS utility function.

 

1.2

Gail for We

8
goes up, and we can see that in (1.4). As we normalize the expenditure level, however,

the interest rate should decrease to offset this tendency as in (1.5).
Also noting E = E' = 1 in (1.3), we get
1

EU) = E'U) = —- (1-6)

2n

1.2.2 Final Goods Manufacturer

Manufacturers of a home originated ﬁnal good, j, uses laborm, Ly(j), and all home

and foreign intermediate goods developed prior to the introduction of the ﬁnal good,11

:r(i, j) and :r(i", j), in their production

Y0) = Ly(j)""[/ x(i.j)°dz' + j: z(i'.j)°di'1”/° (1.7)

where 0 < a,ﬂ < 1. Note that, in each country, there are j intermediate prod-
ucts developed prior to the ﬁnal product j and this is why we have integration of
intermediate goods, i and i‘, over the range [0, j] in the production function above.

Denoting w as the wage rate, perfect competition in the ﬁnal goods sector gives

the demand for labor from producers of the ﬁnal good with the production function

(1.7) as

 

loIn Young’s model, labor is not used in ﬁnal goods manufacturing. Without labor in the ﬁnal
goods sector, however, tariff rates do not affect the growth rate as their effects on patent prices and
wage rates cancel each other.

1’Also in Young’s model, instead of using all previously developed intermediate goods, productions
of ﬁnal goods use only relatively new intermediate products. As mentioned earlier, this change is
only for the simplicity and does not affect the result of the model.

 

1.2..

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of lah

LYU) = _'— (1.8)

Also denoting Pm(-) as the price for an intermediate good faced by home manufac-
turers of ﬁnal goods,” the demands for intermediate products, i and i“, by producers

of a home originated ﬁnal good j are

 

. . _ Pm(i,j)‘/°'“ 1
”(w ’ ‘ I: Pm(i.j)°/°‘"di + I: Pm(i-,j)°'/°-1di" ﬂ 2n (1‘9)
and
._ . P... i‘, ' Va“ 1
2(1 ,J)= ( 1’ ﬁ--—- (1.10)

 

shears-w + f3 enhance-1dr ' 2n

A detailed derivations of the equations above are found in Appendix. Even though
the equations above are for the manufacturers of a home originated ﬁnal good, we can
have the similar equations for the manufacturers of a foreign originated ﬁnal good by

switching j with j". We can also use (1.9) and (1.10) for foreign manufacturers of a

ﬁnal good by changing a: to :r“ and Pm to P,‘,',.

1.2.3 Intermediate Goods

A manufacturer of an intermediate good, who acquires the right to produce the good
from a researcher, can produce one unit of the intermediate good by using one unit

of labor which is paid the wage rate w. Given the demand function (1.9) from home

 

”It could be different from the price intermediate goods producers actually get because, in the
case of the border crossing, it is assessed with tariffs.

 

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manufacturers of a home originated ﬁnal good, a home producer of an intermediate

good then maximizes operating proﬁts

«(231): [Pm(z'.j)— w] - mm) (1.11)

Using (1.9), if we solve the maximization problem of (1.11), we get the mark-up

pricing by a home manufacturer of intermediate goods to home manufacturers of a

ﬁnal good as

Pm(i9j) =

DIS

(1.12)

The mark—up to foreign manufacturers of ﬁnal goods by a home producer of an inter-

mediate good can be derived by maximizing after tariff operating proﬁts

«‘0'.» = 11—}; - P.:.(i.j) — w] - x‘(z‘.j) (1.13)

Using (1.9) with *, if we solve the maximizing problem of (1.13), we get the mark-up
by a home manufacturer of intermediate goods to foreign manufacturers of a ﬁnal

good as

P;(i.j) = 207%) (1.14)

From (1.9), we can see that the demand for a home intermediate product i from
producers of a home manufactured ﬁnal good depends on prices of all home and

foreign intermediate goods developed by the time when the ﬁnal good is introduced.

 

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11
Substituting (1.12) and (1.14)13 in (1.9) we get the demand for a home intermediate

good i from producers of a home ﬁnal good as

a 1 E

309]) = j_w'1+(1+5)a/a-1 . E;

(1.15)

An increase in the tariff rate raises the overall price level of intermediate products
used in the production of a home manufactured ﬁnal good as prices for imported
intermediate products increase. This will help the sales of a home intermediate prod-
uct to producers of a home ﬁnal good as its own price stays the same. In (1.15), the
second fraction with 6 represents this. Also substituting (1.12) and (1.14) in (1.10)

with appropriate changes,M we get the demand for a home intermediate good i from

producers of a foreign ﬁnal good as

_ . . _ a (1-1-6)1/""l ﬂ
$(1a])—jw 1+(1+6)a/a-1 %

(1.16)
As the fraction with 6 in (1.15) represents, the denominator of the fraction with 6
in (1.16) represents the increase in the overall price level of intermediate products
used in the production of a foreign ﬁnal good. The increase in the tariff rate, this
time, raises the prices of all exported intermediate products to the foreign market.
Holding constant the price of the intermediate good under consideration, this increase

in the overall price level helps the sales of the intermediate good in that market. The

increase in the tariff rate, however, also raises the own price and the sales of the good

 

13We actually need Pm(i‘, j) in (1.9). But (1.14) is the symmetric expression for that.
“We need to change 1:(i‘,j) to :r“(i,j), Pm(i,j) to Pg,(i‘,j), and Pm(i‘,j) to P;(i,j)

 

 

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in {1.16

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i aCtua

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11
Substituting (1.12) and (1.14)13 in (1.9) we get the demand for a home intermediate

good i from producers of a home ﬁnal good as

a 1 ﬂ

3(3,]) = 3; . 1+ (1 +6)0/a-1 . a;

(1.15)

An increase in the tariff rate raises the overall price level of intermediate products
used in the production of a home manufactured ﬁnal good as prices for imported
intermediate products increase. This will help the sales of a home intermediate prod-
uct to producers of a home ﬁnal good as its own price stays the same. In (1.15), the
second fraction with 6 represents this. Also substituting (1.12) and (1.14) in (1.10)

with appropriate changes,” we get the demand for a home intermediate good i from

producers of a foreign ﬁnal good as

g (1 +<5)"°"1 ﬂ

1' (2’)) = jw . 1+(I +6)a/a-l . '27;

 

(1.16)

As the fraction with 6 in (1.15) represents, the denominator of the fraction with 6
in (1.16) represents the increase in the overall price level of intermediate products
used in the production of a foreign ﬁnal good. The increase in the tariff rate, this
time, raises the prices of all exported intermediate products to the foreign market.
Holding constant the price of the intermediate good under consideration, this increase
in the overall price level helps the sales of the intermediate good in that market. The

increase in the tariff rate, however, also raises the own price and the sales of the good

 

13We actually need Pm(i',j) in (1.9). But (1.14) is the symmetric expression for that.
1‘We need to change z(i‘,j) to z'(i,j), Pm(i,j) to P;,(i',j), and Pm(i',j) to P;,(i,j)

12

under consideration as it is exported to the foreign country. The numerator in the
fraction with 6 in (1.16) represents this negative direct price effect.

The ith home intermediate product would be used in the production of all home
' and foreign originated ﬁnal products introduced later than the intermediate product.

As each type of ﬁnal product is produced in both countries, the overall demand for

the intermediate good i can be found as15

use) = 2]"1x(i.j) + x‘wnatj

Substituting (1.15) and (1.16), we have

2(i) = i—i - ¢(6)In(%) (1.17)

where

1+(1+.5)1/°'-1 1 8
1+(1+.5)o/or-1 ( -‘ )

 

¢(5) 5

Because 41(6) is crucial to explain the movement of the equilibrium growth rates later,
it is important to note that the function is initially decreasing and then increasing
as it asymptotically approaches the initial level as the tariff rate, 6, increases from
zero to inﬁnity}? As we can see from (1.15) and (1.16), 43(6) is the sum of the two

terms representing sales of intermediate good to producers of ﬁnal goods in both

 

l"’As Young’s model has an upper bound of intermediate goods used in the production of a ﬁnal
good, it is integrated over j E [i/Q, n] there.
1643(6) here is equivalent to f(r) in Rivera-Batiz and Romer [1991].

 

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countries. An increase in the tariff rate raises the overall price level and positively
affects the sales of the intermediate good in consideration in both countries. However,
the increase in the tariff rate also raises the own price of the intermediate good and
decreases its sale in the foreign market. With the tariff rate close to zero, this negative
effect dominates the positive overall price effect. This explains the initial decrease
in the 45(6). As the tariff rate is increased further, however, the dominance of the
negative direct price effect for sales in the foreign market weakens. As the tariff rate
approaches inﬁnity, sales in the foreign market approaches zero. However, the sales
in the home market becomes twice as much as the free trade level. This explains why
the ¢(6) function goes back to the initial level as 6 approaches inﬁnity.

The overall proﬁts for the producer of the intermediate good i can be found as

«(2‘) = 2 ("mm + w'(i.j)1dj

By substituting (1.15) into (1.11 ), (1.16) into (1.13) and those results into the above,

we have”

«(2) = (3%)}: - ¢(6)In(’:—.) (1.19)

By differentiating (1.19)with respect to n, we get

370) _ (1 - 0W4“) (1 - a)ﬂ¢(5) 3
E... _ — In( i ) (1.20)

n2 n2

 

 

17Even though we omitted time variable t here, it is the operating proﬁt for the intermediate good
i at time t.

14

Because a new ﬁnal product uses all existing intermediate products, it will increase
the demand for each intermediate good developed earlier than the ﬁnal product.
This is the complementarity effect and the ﬁrst term on the right hand side of (1.20)
represents this. As the new ﬁnal product is introduced, however, consumers have to
divide their given level of expenditures over a wider variety of ﬁnal products. This
will reduce their demand for each ﬁnal product and thus decrease the demand for
each intermediate product. This is the substitution effect and the second term on the

right hand side of (1.20) represents this.

1.2.4 Research

Researchers in the home country produce new designs for intermediate products ac-

cording to18

2L3“) ‘ "(U

a

n’(t) =

 

(1.21)

With free entry into the research sector, the unit cost of a new design for this technol-
ogy, wa/ 2n, equals the price of the nth patent design, PR[n(t)], which is the present

value of the operating proﬁts that will accrue to the manufacturer of the intermediate

product.

= w“ = = °° -[R(s)—Ru)1.
UC_2n(t) 133-]! e 7r[n(t),s]ds (1.22)

 

 

1“There is no restrictions on the ﬂow of ideas among researchers worldwide as they can use all
previous knowledge in their research.

15

where the equality in the middle holds when there is positive growth.

By substituting (1.19) for the intermediate good n(t) at time 3 into (1.22), we

have

 

wa = °°(1-a)ﬂ ”ﬂ e-{Rm-Rm] 3
2n(t) j: 71(8) ¢(6)I ['10)] d (1.23)

1.2.5 Labor Market

To close the model, we need to describe the labor market. Labor demand in the
research sector can be found from (1.21) as

an’ ag

Lam = X = 3 (1.24)

Labor demand in the ﬁnal goods manufacturing sector can be found by using (1.8)

as

1-H

w

Ly(t) = 2 /" Lymdj : (1.25)

As one unit of labor is required for a unit production of an intermediate good, labor

demand in the intermediate goods manufacturing sector can be found by using (1.17)

as

LM(t) = [Sandi = “—ﬁE—m /0"zn(%)ﬁ = M (1.26)

w

where the last equality holds as fun In( %)di 2 n. Denoting L = L" as the total labor

16
endowment in each country, we get the labor market clearing condition by combining

(1.24), (1.25) and (1.26) as

g_g_ 1—3+a3¢(6)

 

 

2 'l' w w = L (1.27)
By substituting (1.27) into (1.23)

It is useful to note from (1.5) that along the equilibrium path, we have e‘[R(‘)'R(‘)l =
e"("’)- efo‘gmd". It is also useful to note In[n(s)/n(t)] = f,’ g(v)dv. Using these, we

have

0345(5)+

g_r_._
6

5i ﬂ = (1- 0WW” £m[£.9(v)dvle‘”(""ds (L29)

1.3 Steady State Equilibria

In a steady state equilibrium, we will have a constant growth rate 9. To ﬁnd the

steady state value of 9, we integrate the last part of (1.29) for a constant value of g

and get

00 3 d _p(s-¢) = i .
f1 {/tgv]e d3 p2 (130)

Substituting (1.30) into (1.29), we have

17

 

05¢(5)+1- B _ (1 - Oil/345(5)!)
3% - g - p’

(1.31)
If we multiply the left and the right hand sides of (1.31) by 1 / n they become, respec-

tively, the unit cost and the patent price of a new product design.

6 1— 1
110:“‘34’2; 5.; (1.32)

and

Pa = (1 - 0225¢(5)g

I
- H (1.33)
It is useful to note that they are both increasing in 9. With the ﬁxed labor endowment
in the economy, an increase in research activities puts upward pressure on the wage
rate as we see in (1.27) and thus raises the unit cost of a patent design. As the
growth rate increases, the number of ﬁnal products that use an intermediate product
grows faster and the price of a patent design increases because the patent price is the
present value of a flow of operating proﬁts.19 Figure 1.1 below represents the unit

cost and the patent price against the growth rate (g) and equilibrium growth rates

are represented by the intersections of two curves. Denoting

K ___ pzlaﬂw) + 1 — ﬂ]

" (1 mums) “'34)

 

19It is linearly increasing in g as the effect of g on the interest rate and on the substitution effect
cancels out as we see in the transition of (1.28) to (1.29).

18

if g < x/Tf, the two curves will never intersect and there will be no steady state
equilibrium with positive growth. If % = m, the two curves will touch each other
once and the tangent point, g = f, will be the only steady state equilibrium with
positive growth. However, rather than analyzing this knife edge case, our attention
will be on the case where I: > m, in which the two curves intersect twice and we
have two equilibria with positive growth. As we see in the Figure 1.1, the price of the
patent is increasing at a constant rate while the unit cost of the new patent design
is increasing at an increasing rate. Holding n constant, if we differentiate (1.32) and

(1.33) with respect to g and divide those results by the originals, we get

 

UC’ g 1
% = 217:; (1-35)
and
’ 1
if” = 5 (1.36)

From (1.35), we can see that the rate at which the unit cost of a new design
increases is increasing with the growth rate. As we can see in (1.24), an increase
in the growth rate directly increases the demand for labor in the research sector.
Holding demand for labor from manufacturing sectors constant, this increase in the
demand for labor in the research sector raises the equilibrium wage rate. At a lower
growth rate, however, employment in the research sector is relatively small compared
to that in manufacturing sectors. Thus, additional demand for employment in the

research sector due to increases in the growth rate will not put much pressure on the

19

UC P

 

 

“Q """
fl
1o

Figure 1.1: U0 and PR

wage rate and thus the unit cost to increase. As the growth rate further increases
and the portion of labor employed in the research sector becomes signiﬁcant, however,
additional increases in the growth rate and in the demand for labor from the research
sector will put greater pressure on the wage rate. This will signiﬁcantly raise the unit
cost of developing new products. Thus, there exists a positive relationship between
the growth rate and the speed at which unit cost increases.

From (1.20), we can see that the complementarity and the substitution effects of
new product development affect the instantaneous operating proﬁts in opposite direc-
tions. As mentioned earlier, since the patent price is the present value of the ﬂow of
this operating proﬁts, we can see that the increase in the growth rate and thus that
in the number of products have two conﬂicting effects on the patent price. When i is
relatively close to n (i.e. when the intermediate good i is a relatively new product),

the complementarity effect dominates the substitution effect. As 12 grows further,

20

however, the substitution effect becomes dominant. If the growth rate is slow, a new
product will enjoy a long period of complementarity dominance and the effect of the
growth rate on the patent price through complementarity will be relatively large. As
the growth rate increases, however, the period of complementarity dominance will be
shortened and the effect of the growth rate on the patent price through the comple-
mentarity will be relatively small. Thus, slower growth corresponds to faster patent
price increases and faster growth corresponds to the slower patent price increases, as
shown in (1.36). At lower growth rates, the patent price grows faster than the unit
cost does, and later, at higher levels of growth , the speed at which these two increase
is reversed. In fact, (1.35) and (1.36) indicate that the patent price is increasing at a
higher rate than the unit cost when 9 is less than %, and vice versa.

In Figure 1.1, two equilibria with a positive growth rate occur at points where the
unit cost and the patent price are equal. The equilibrium actually achieved by the
economy depends on the expectations of researchers. If they are pessimistic about
the rate of future research activities, the economy will settle down on a lower growth
rate equilibrium. On the other hand, if they expect a higher rate of the research
activities in the future, the economy will attain a higher growth rate equilibrium.
The equilibrium with the growth rate less than L / a is the complementarity dominant
equilibrium as it occurs where the complementarity effect is relatively larger. Similarly
the equilibrium with the growth rate higher than L / a is the substitution dominant
equilibrium as it occurs where the substitution effect is relatively larger. The two

equilibria can be algebraically obtained by solving (1.31) and they are

9c: %- “(ﬁr-K (1.37)

and

g.=-:1+[/(%)’ - K (1.38)

There are issues related to the fact that only the complementarity dominant equi-
librium is stable in this model. However, as Young points out, the stability problem
should not necessarily lead one to conclude that the complementarity dominant equi-
librium is the most likely long-run outcome. In models with learning environment,20
different assumptions about the informational structure lead to quite different sto-
ries about stability. In fact, using a more general form of the CES utility function,
we could construct a model in which both the complementarity and the substitution
dominant equilibria are stable. In the analysis below, however, we will bypass these

stability issues and assume that both equilibria could be achieved through appropriate

processes.21

1.4 Growth Effects of Tariffs

For a comparative static analysis of the effects of tariffs on both equilibrium growth

rates, we need to examine (1.37) and (1.38). However, as K in (1.37) and (1.38) is

 

”See Grandmont [1985] and Lucas [1986].

21For example, once off the substitution dominant equilibrium, we could jump back to the new
substitution dominant equilibrium.

22
the only term that contains the tariff rate(6), we simply need to differentiate (1.34)

with respect to 6 to get

{11$__p’(1 -ﬂ). 49(5)
66 " (1 -a)ﬂ 1416)]? (1'39)

Recall from (1.18) and the explanation after that, the 45(6) function is ﬁrst decreasing
and then increasing as it asymptotically approaches the initial level. If we take a
look at the complementarity dominant equilibrium ﬁrst, as there are total of three
negative signs at the beginning of the right hand side of (1.39) and in front of K and
the square root sign in (1.37), we can infer the movement in the equilibrium growth
rate. As shown in Figure 1.2 below, an initial increase in the tariff rate from 6 = 0
increases the complementarity dominant equilibrium growth rate. The effect will be
reversed at higher tariff rates, and, as the tariff rate goes to inﬁnity, the growth rate

will asymptotically approach the zero tariff growth level.

 

 

 

 

Figure 1.2: Changes in the Growth Rate

23

Since the signs in front of the square roots of both equilibrium growth rates in (1.37)
and (1.38) are the only difference between them, tariff rates have exactly opposite
effects on the substitution dominant equilibrium growth rate.

To get an insight into the movements in equilibrium growth rates, we need to

examine the unit cost and the patent price in (1.32) and (1.33). Differentiating them

with respect to the tariff rate (6), we get

 

aUC' __ (1)8 ,

86 — (2.5. -g)n - 42(6) (1.40)
and

aPR_(l-a)ﬂg. I

—06 _ ___”: ¢(5) (1.41)

As we note in (1.40) and (1.41), both have the term (6’ (6) and this indicates that both
U C and PR move in the same direction as 65(6). That is, both the unit cost and the
patent price decrease as the tariff rate increases at a lower level and then increase at
a higher tariff level.

Figure 1.3 shows how the U C and PR curves respond to the changes in the tariff
rate. Starting from a tariff rate of zero, both curves shift to the right as the tariff
rates increases, indicating that both the unit cost and the patent price are decreasing.
Above a certain level of the tariff , however, both curves start to move back and
asymptotically approach their original positions.

Recall that the patent price is the present value of the proﬁt ﬂows for intermediate

good manufacturers and depends on sales of the product at home and abroad. The

24

  
    

 

 

“'9
a"!
3
‘9
1a

Figure 1.3: Effects of tariff on U C and PR

sales in both markets, in turn, depend on the tariff rate. As we explained after the
introduction of 45(6) function in (1.18), the sales and thus the patent price of a product
decrease at lower levels of the tariff rate and then at higher levels of the tariff rate
increase back to the level associated with zero tariff rate. This is why the PR curve
in Figure 1.3 ﬁrst shifts to the right and then back to its original position.

An increase in tariff rates at lower levels decreases the demand for intermediate
goods, as we noted earlier. Since one unit of labor is required for a unit of intermediate
goods, this reduces the demand for labor from the intermediate goods sector. Holding
labor demand in other sectors constant, the released labor from the research sector
decreases the wage rate and thus lowers the unit cost of developing a patent design.
At a higher tariff rate, however, the demand for intermediate goods and for the labor
from the intermediate goods manufacturing sector bounce back as increasing domestic

sales outweigh decreasing foreign sales. Therefore, the U C curve in Figure 1.3 also

25
initially shifts to the right and then goes back to its original position as the tariff

rates increase.

If we divide (1.40) by (1.32) and (1.41) by (1.33), we get

U 0(5) _ 0546(5) . ¢’(5)

U0 " 1 — ﬂ + aﬂ¢(6) 4(6) “‘42)

and

{2(5) _ 2m
PR - ¢(5) (1.43)

As 0 < 6 < 1, the ﬁrst fraction on the right hand side of (1.42) is less than 1 and,
together with (1.43), this indicates that the rate at which the patent price is changing
is greater than that of the unit cost, whatever direction they are moving. The effects
of a tariff on the patent price is more direct than its effects on the unit cost of a newly
developed product. This is because the patent price depends on the sales and proﬁts
of the intermediate goods producer and the tariff rate affects them directly, while
tariff rates directly affect the intermediate goods manufacturers’ demand for labor,
which is only a part of the labor market. This change in the labor demand indirectly
affects the wage rate, and thus the unit cost of a product development. In whatever
direction they are moving, therefore, the PR curve shifts more than the U C curve.
At lower levels of the tariff rate, where the two curves are shifting to the right,
(1.35) and (1.36) along with (1.42) and (1.43) indicate that an increase in the tariff
rate moves any equilibrium growth rate below 9 = L / a to the right and any equilib-

rium growth rate above g = L / a moves to the left. At higher levels of the tariff rate,

26

where both curves are shifting to the left, the movements of the equilibrium growth
rate are reversed as both curves asymptotically reapproach their original positions.
Thus, we will have movements of the equilibrium growth rates in response to the
change in the tariff rate as in Figure 1.2.

We can ﬁnd the intuition behind the movement of the equilibrium growth rates by
investigating equations related to the labor market. Holding the wage rate constant,
we can see the initial decrease in the demand for labor from the intermediate goods

manufacturing sector by differentiating (1.26) with respect to 6

312M _ 0305'“)
E’L - —w (1.44)

As the ¢' (6) term, included in the numerator of (1.44), ﬁrst decreases and then in-
creases, we can see that the demand for labor in the manufacturing of intermediate
goods is initially decreasing. This decrease in labor demand lowers the wage rate.

This can be seen by differentiating (1.27) with respect to 6 while holding Ly and LR

constant,

0w _ aﬂ¢’(6)
(96 - _ L — ag/2

YoUR
This decrease in the wage rate increases the demand for labor from the intermediate
goods sector, but not enough to overcome the initial decrease. Thus, the overall effect
of the increase in the tariff rate at lower levels will be to decrease the employment of

labor in the intermediate goods sector.

This decrease in the wage rate will also affect the demand for labor from both

27

the ﬁnal goods manufacturing sector and research sector. As tariff rates increase at a
lower level, the value of marginal product of labor in both the research sector and the
ﬁnal goods manufacturing sector decreases. Holding the labor employment in each
sector constant, if we differentiate the values of marginal product of labor in both the

research and the ﬁnal goods manufacturing sectors, we have

BVMPLR _ 1166(6)
56' _ L — ag/2 (1.45)

and

06 L — ag/2 L - ag/2 (1°46)

As perfect competition prevails in both the research and the ﬁnal goods manufacturing
sectors, decreases in the value of the marginal product of labor in both sectors tends
to reduce the demand for labor in both sectors.

Therefore, an increase in tariff rates at lower levels has a tendency to release
labor from all three sectors. If we consider the tendency of releasing labor from the
intermediate and the ﬁnal goods manufacturing sectors together as opposing forces
against the tendency of releasing labor from the research sector, the relative force
releasing labor from the manufacturing sector outweighs that from the research sector
at the complementarity dominant equilibrium. Thus labor in the research sector and
the complementarity dominant growth rate actually increase as tariff rates increase
from zero. At the substitution dominant equilibrium, the relative sizes of forces out

of each sector are reversed and the equilibrium growth rate initially decreases as the

28

tariff rate increases.22

The above analysis considers only the case of lower levels of the tariff rate. Above
a certain level, a further increase in the tariff rate will lower the complementarity
dominant growth rate and will raise the substitution dominant growth rate as each
sector tends to attract labor. Furthermore, as the tariff rate goes to inﬁnity, the
employment level in each sector and the growth rate go back to the initial free trade
levels. As we explained in the movement of 45(6) function in (1.18), this is because
that the sales of an intermediate good to producers of ﬁnal goods in the home country
doubles while the sales in the foreign country becomes zero. This makes the patent

price and the unit cost of a design for a new intermediate good go back to the original

level associate with free trade.

1.5 Comparison with Rivera-Batiz and Romer

In Rivera-Batiz and Romer, tariffs alter the allocation of resources (human capital)
between the manufacturing and research sectors which in turn affects growth. In
their model, human capital is used in both manufacturing of ﬁnal goods and research
activities. As the tariff rate increases, the flow of intermediate goods between two

countries is hindered. In manufacturing of ﬁnal goods, producers use less imported

 

”In this model, the critical value for the growth rate at which the relative size of forces releasing
labor is reversed is g = L/a. Substituting this into the R660 labor employment condition (1.24),
we get the critical value for the employment of labor in the research sector as L/2. Then, the R&D
labor employment in the substitution dominant equilibrium is more than 50% of the population and
this is obviously unrealistic. However, as in Young, by changing the range of intermediate goods
used in the production of a ﬁnal good from [i, n] to [i/0, n] so that there is an upper limit for the
index of intermediate goods that can be used in the production of advanced ﬁnal products, we can
have the substitution dominant equilibrium with labor in the research sector less than L / 2.

29

intermediate inputs. This will lower the marginal product of. human capital in the
manufacturing sector. The fact that manufacturing of ﬁnal goods uses less imported
material means that the manufacturers of intermediate goods export less. This de-
crease in sales abroad results in a decline in patent prices, which equal the present
value of operating proﬁt flows for intermediate goods producers. This decrease in the
patent price, in turn, decreases the value of the marginal product of human capital
in the research sector.

As patent price is increasing linearly with the quantity of intermediate goods sold,
and is not a function of anything else, the increase in the tariff rate directly affects the
marginal product of human capital in the research sector. On the contrary, the effects
of tariff increases in manufacturing is indirect in the sense that increases in the tariff
rate only affect the imported intermediate goods which must be combined with other
inputs, namely, labor, human capital, and domestically produced intermediate goods.
Thus, with tariff rates close to zero, we have decreases in the value of marginal product
of human capital in the research sector greater than that in the manufacturing sector.
This causes human capital to move from the research sector to the manufacturing
sector and, consequently, the growth rate to decrease. As tariff rates keep on growing,
the marginal product of imported intermediate goods keeps rising because fewer of
them are used in production. Thus at higher tariff rates, the decrease in usage of
imported intermediate goods will cause a large enough reduction in the marginal
product of human capital in manufacturing to outweigh that in the research sector
and the trends in resource allocation is reversed. Therefore, the increase in tariff rates

at higher levels increases the growth rate.

30

In our model, the source of growth effect of tariff changes is also the allocation
of resources among sectors. The resource here is labor instead of human capital.
Furthermore, effects of a tariff on the value of the marginal product of labor in both
research and manufacturing of ﬁnal goods sectors are the same as that on the value
of marginal product of human capital in the Rivera-Batiz and Romer model. In fact,
if we divide (1.45) by the value of marginal product of labor in the research sector
and (1.46) by that in the ﬁnal good sector, we can show that the force moving labor
in and out of the research sector is always stronger than that in and out of the ﬁnal
goods manufacturing sector. However, contrary to human capital being used only
in the research and ﬁnal goods manufacturing sectors in their model, labor is used
in all sectors here, including the intermediate goods sector. Therefore, we will have
an additional source of resource allocation in our model, namely, the flow of labor
in and out of the intermediate goods sector. At lower tariff rates, labor moves out
of the intermediate goods sector. At the complementarity dominant equilibrium, the
combined labor flow out of manufacturing of both intermediate and ﬁnal goods over-
whelms the labor ﬂow out of the research sector while the magnitudes are reversed at
the substitution dominant equilibrium. Thus, we will have increasing complementar-
ity dominant and decreasing substitution dominant growth rates at the lower tariff

rates, as explained in the previous section.

1.6 Conclusion

Effects of the change in the tariff rate on the equilibrium growth rate in this model
crucially depend on pe0ple’s expectations about the future. With the tariff rate low
enough, if the economy is currently in the substitution dominant equilibrium where
researchers are expecting a higher rate of invention, the decrease in the tariff rate
raises the common growth rate. This effect of the tariff reduction on the substitu-
tion dominant equilibrium is in line with the last four decades of GATT’s continuing
efforts of reducing tariffs among countries. On the other hand, if researchers are
expecting a lower rate of invention so that the economy is in a complementarity
dominant equilibrium, a mutual reduction in the common tariff rate decreases the
growth rate in both countries. This may explain the productivity slow down in late
19703 and early 19803. As is well known, it was a period of uncertainty caused by
a huge increase in oil prices. This may cause innovators become pessimistic about
the future innovation and economic situations might become the one which is suit-
ably explained by complementarity dominant equilibrium. If, in fact, the economy
had been in complementarity dominant equilibrium at that time, the continual tariff
reductions by GATT agreements may reduces rather than enhance the world growth
rate. Even though, it must be empirically answered whether the economy was at the
complementarity or the substitution dominant equilibrium, the analysis here suggests
that care should be taken in implementing the trade policy with a model. We may
have to make sure that a certain peculiar situation in an economy does not lead to an

exceptional direction away from the conventional wisdom of the free trade advocate.

31

32

Our conclusions may depend on the structure of our model and the assumptions
we are made. As mentioned earlier, for example, if we ad0pt different structures
for learning environment, we could have different conclusions about the equilibrium
selection. As the name suggests, new growth theory itself is quite new and many areas
of research remain. Further exploration of the model presented here is necessary in

order to determine how robust our results are.

Chapter 2

A R&D base Growth Model with

Human Capital Accumulation

2.1 Introduction

One of the problems found in many ﬁrst generation endogenous growth models is the
scale effect. That is, all the models, developed by Romer [1990], Segerstrom, Anant
and Dinopoulos [1990], Grossman and Helpman [1991], and Aghion and Howitt [1992],
predict a faster economic growth with larger labor force in the R&D sector. However,
this scale effect is not well reﬂected in data. The NSF [1989] data found dramatic
increases in the number of scientists and engineers engaged in R&D in the US. for
the last four decades. If the prediction in these models is right, we should also have
dramatic increases in the growth rate for the US. economy. However, as Jones [1995a]

argues, the average per capita growth rate has been either constant or declined since

1950.
33

34

By making the marginal size of innovations decreases as the total stock of knowl-
edge increases, Jones [1995b] successfully remove the scale effects from earlier versions
of R&D based endogenous growth models. However, as only exogenously determined
parameter values like the population growth rate affect the long-run economic growth
rate, Jones’ model has a rather strong implication. Segerstrom [1995] also develops
an endogenous growth model which is free of the scale effects. In a quality ladder
setting, in which innovators are trying to improve the quality of existing products, he
adds an endogenous component to economic growth. That is, with or without popu-
lation growth, the endogenously determined human capital growth rate can affect the
long-run economic growth rate. He also shows that education subsidies can increase
the long-run economic growth rate.

This paper presents a model of horizontal product differentiation, as in Jones,
while incorporating human capital accumulation as in Segerstrom.1 The long-run
equilibrium economic growth rate is determined by the combination of the endoge-
nously determined human capital growth rate and the exogenously given population
growth rate. We ﬁnd that an improvement in the efficiency of human capital ac-
cumulation can affect the long-run economic growth rate. Especially, we ﬁnd that
long-run growth effects depend on the elasticity of substitution. In fact, in the model
presented here, when consumption at two different points in time are highly substi-
tutable, increases in the productivity of human capital accumulation lead to slower

economic growth. We also ﬁnd that the effect of increases in the population growth

 

lHuman capital in this paper is different from that in Segerstrom, as human capital and labor
are separate inputs.

35
rate on human capital accumulation and the long-run economic growth also depends
on the elasticity of substitution.

In Section 2 below, the structure of the model is described. The solution of the
model for steady state equilibria with positive growth rates is developed in Section
3. Then in Section 4, the comparative dynamic analysis is presented. Section 5 sum-
marizes the ﬁndings of this paper and suggestions for future research are discussed.

Finally, in the appendix, the solution for the optimal control problem of consumers

is presented.

2.2 The Model

Researchers in this economy try to come up with new designs for intermediate goods
which are different from existing intermediate goods. Once successful in developing a
new good, the researcher is awarded with an inﬁnite patent protection, and then, sells
its production right to the producer who submits the highest bid. With an inﬁnite
number of producers who want to produce the product, the equilibrium price for the
patent design equals the present value of proﬁts that will accrue to the producer of
the intermediate good. As the producer has the exclusive right for production, he
can charge producers of the ﬁnal good the monopoly price for the intermediate good.
Final good producers use all the existing intermediate goods along with labor and
human capital in their production. The ﬁnal good sector is characterized by perfect

competition. Finally, consumers maximize their utility by consuming this ﬁnal good.

36

2.2.1 Consumers

Denoting v(t) as the fraction of human capital at time t devoted to its accumulation,

we assume that consumers in this model accumulate human capital according to2

H(t) = AH(t)[1 — v(t)] (2.1)

where A > 0 is a given technology parameter that measures how effective workers are
in accumulating human capital. Human capital that are not devoted to its accumu-
lation (2211) will be employed in either research (H .4) or manufacturing of ﬁnal good
(Hy).

A representative consumer who has the subjective discount rate p > 0 and con-

sumes c(t) at timet maximizes

loco e"”u(q)dt (2.2)

subject to a dynamic budget constraint and the differential equation for the human

capital accumulation given in (2.1). Assuming that the utility function, u(-), in (2.2)

 

2A similar speciﬁcation is used in Lucas [1988] and Segerstrom [1995] so that workers can maintain

a constant rate of human capital growth over time by devoting a constant fraction of their time to
studying.

37

is a Ramsey type preference with constant elasticity of substitution equal to 1 / d > 0,3

 

de=

if we solve the consumer’s optimal control problem as in Appendix, we get two in-

tertemporal optimization conditions. The ﬁrst one is for consumption growth,

0-p-n) (1%

am-
QI"

where r is the interest rate and n is the population growth rate. It is important to note
that consumption growth is increasing in the consumer’s elasticity of substitution.
Consider consumption at two different points in time. If the interest rate (r) is
raised, consumption at the earlier point in time becomes more expensive relative to
consumption at the later point in time. Consumers will then tend to postpone their
current consumption in return for more consumption in the future. With a higher
elasticity of substitution, that is, with a lower value of d, it will be easier to postpone
the current consumption and the growth rate in consumption will be higher.

The second intertemporal optimization condition is related to human capital ac-

cumulation.

 

3In Jones. u(c.) = Eli-WV; is said to exhibit the constant relative risk aversion of l/d. However,
if the von Neumann-Morgenstern utility integral is additively separable, both the elasticity of sub-
stitution and the relative risk aversion depend only on the curvature of the instantaneous utility
function and thus two concepts are inter-changeable. See chapters 2 and 6 of Blanchard and Fischer
[1989] for detailed discussions. Since there is no uncertainty involved in consumers behavior in this
model, it is more appropriate to use the term of the constant elasticity of substitution.

38

A+—=r (2-4)

where w), is the wage rate for human capital. As the technology parameter A measures
the marginal increase in growth of human capital for a given increase in the fraction
of human capital devoted to its accumulation, (2.4) requires that the marginal rate

of increase in human capital plus the growth rate in the wage for human capital must

equal the market interest rate.

2.2.2 Manufacturing

Final Good

Using human capital (Hy), labor (L), and all intermediate goods developed (:r(i)
where i E [0, A]), the ﬁnal good sector produces a consumption good, Y, according

to constant returns to scale technology:

A 0

Y = HeLﬂ / :1:(i)1‘°"'f’dz o < as < 1 (2.5)

0
Note that intermediate goods enter into the production of ﬁnal good in the form of
horizontal differentiation as speciﬁed in Ethier [1982] and Dixit and Stiglitz [1977].
Innovation in this model is represented by increasing variety of these intermediate
goods (i.e. increase in A). Normalizing the price of the ﬁnal good to unity, perfect

competition in the ﬁnal good sector yields conditional demand functions:

39

Hy = a- L (2 6)
wk
and
p(i) = (1 — a - ,3)H$Lf’:r(i)‘(°+5) vi (2.7)

where p(i) is the rental price of intermediate good 2'.

Intermediate Goods

As speciﬁed in Romer [1990] and Jones [1995b], we assume that a producer of an
intermediate good who acquires a patent design from a researcher rents a consumption
good‘ at a unit rental rate r for a period and transform it into a single unit of the
producer durable. The produced intermediate good is then rented out to ﬁnal good
manufacturers at a rate p(i). At the end of the period, the producer durable is
transformed back into the original form without any depreciation. The manufacturer

of an intermediate good then tries to maximize the operating proﬁt:

«(5) = p(i).r(i) — r . 3(2) (2.8)

Using (2.7) in (2.8), we get the mark-up price for all intermediate goods as

 

4This is the same as saying that there exists a separate sector producing capital goods in which

they uses the same technology as in ﬁnal goods sector. In either case, the capital is the forgone
consumption.

40

190') = p = (2-9)

l—a—ﬂ
Noting the symmetry of the mark-up price for all i in (2.7) and (2.8), we can also note
the symmetry in demands and in proﬁts for all existing intermediate products. This

symmetry in demands for each intermediate good especially gives us a convenient way

of expressing the instantaneous capital stock at a point in time as

A
K = [o xdi = A1: (2.10)

Using (2.10) with (2.9) and (2.7), we can express the interest rate as

r=(1-a—)6)2- (2.11)

XIV

2.2.3 Research

By using human capital, researchers try to design a new type of intermediate product

based on all previously developed designs as

A = 6.4612,, (2.12)

where 6 > 0 is a technology parameter for the research sector and O < 0 < 1 is an
externality parameter that measures the degree of externality across time in the R&D

process.5 While researchers freely use previous knowledge in their research. they have

 

sln Jones, it just assumed that ¢ < 1 so that his model can include the negative externality case,
if < 0. However, as this decreasing return to scale feature is not essential to removing scale effects in

41

to pay for the human capital. As there are no restrictions for entry into the research
sector, the wage rate for human capital must equal the value of the marginal product

of human capital in the research sector. Thus,

in), = PAdAd’ (2.13)

where PA is the price for a patent design. As there are an inﬁnite number of man-
ufacturers who want to produce a patented intermediate good, the patent price will

be the sum of the present value of operating proﬁts which accrue to the producer of

the intermediate good.

PA = j: e' I; '(‘)d‘1r(t)dt

Differentiating this and dividing the result by the original form, we get the usual no

arbitrage condition as

71’ PA
T): + E (2.14)

r:

2.3 Steady State Equilibrium

Noting that a constancy of per capita consumption growth in a balanced equilibrium
path in (2.3), we can see the constancy of the interest rate along a balanced equilib-

rium path. Using this, if we differentiate both sides of (2.11) with respect to time

 

his model, the model presented here sticks with the increasing return to scale assumption of 43 > 0.

42

and divide the result by (2.11) itself, we get the equality between the growth rates of
Y and K in a steady state. Let C E cL denote the total consumption by the whole
population, we can write total physical capital accumulation as K = Y — C. Dividing
both sides of this by K, differentiating both sides with respect to time, and noting
the constancy of K / K in a steady state, we get

_L
Y—C

_C_
Y-C

~<| ~<-
01 c:-
a1 a-

Noting the equality between the growth rates of Y and K in a steady state, we can
also establish the equality between growth rates of Y and C. Thus, denoting g, as

the steady state growth rate of the placeholder x, we can show that

91! = HR = 9c (2.15)

Rewriting (2.12) as A/A = 6Af‘1HA, noting the constancy of A/A in a steady
state, differentiating both sides with respect to time, and dividing the result by the

original form, we get

1 1
9.4 = mgHA = my” (2.16)

where the second equality is obtained by noting the constancy of a steady state share
of human capital devoted to the research sector, which we will derive at the end of

this section.

Differentiating both sides of (2.10) with respect to time, dividing the result by

43

(2.10), and noting (2.15), we get

91' = 9.4 'l" g, (2.17)

Noting the constancy of the rental price for intermediate goods in (2.7), if we differ-

entiate both sides of the equation with respect to time and divide the result by (2.7),

we get

a + [3

9’: 0+6” a+ﬂn

Substituting this into (2.17), we get

a B
9Y=94+—grr+——n

a + 6 a + (3
Noting (2.16), we can rewrite the above as
1 a 3
gr - (m + mlyu + mu (2.18)

From (2.18), we can see that long-run growth of this economy comes from two sources.
They are growth in the stock of human capital and growth in the population. Growth
in the population raises the long-run economic growth as increased p0pulation goes to
the production of ﬁnal goods. While all of the increase in the population is absorbed
in the ﬁnal goods sector, the increased stock of human capital will be split into
two sectors. They are the research and the ﬁnal good sectors. Thus the growth in
the stock of human capital contributes economic growth through two channels. The

ﬁrst term in the parenthesis of (2.18) represents the contribution of human capital

44

increases to economic growth through the creation of new types of products from
the research sector. The second term represents the contribution of human capital

increase through the direct effect of increased human capital employment in the ﬁnal

good sector. From (2.6), we get

gm). =gY -93

Substituting (2.18) into the above, we get

1_3
1—¢ 0+3

)9}! + in

(1+3

 

9...=(

Substituting this into (2.4), we get the interest rate required from consumer’s in-

tertemporal optimization for human capital accumulation as

6
0+3

1 (3
A+(—-$ua+ﬂ

 

 

)9” 'l" n = 7‘ (2.19)

As the term inside the parenthesis is greater than zero, we can see that there is a
positive relationship between the growth rate of human capital and the interest rate.
Noting that the last two terms in the left hand side represent the growth rate of the
wage for human capital, if we have an increase in the interest rate, we need to have an
increase in the growth rate of the wage for human capital. As the ﬁnal good sector is
perfectly competitive, the growth rate of the wage for human capital must equal the
growth rate of the marginal product of human capital in the ﬁnal good sector. The

growth rate of the marginal product of human capital is the difference between the

45

output growth rate and the human capital growth rate as indicated by 9.1,, = 9,, — go
above. It seems as though the human capital growth negatively affects the growth
rate of the wage for human capital. As we noted earlier in (2.18), however, growth in
ﬁnal good production also comes from human capital growth, as well as population
growth. As growth in human capital affects the ﬁnal production not only by the
increase in employment of human capital but also by the increase in the number of
intermediate products through R&D, the overall effect of an increase in the growth
rate of human capital on that of the wage rate for human capital will be positive.
Thus, we have a positive relationship between the interest rate and the human capital
growth rate. We can represent this relationship as the HH curve in the r - 9,, plane

in Figure 2.1 below.

From the deﬁnition C 5 CL, we get C / C = é/c + n. Substituting from (2.3), we

get

1
9c = ;(r—p-n)+n (2.20)

Substituting (2.20) into (2.18) and noting (2.15), we get the interest rate required

from consumers balanced expenditure growth as

1 a a
d[(1_¢+a+3)g”—a+ﬂn]+p+n—r (2.21)

 

 

 

Note that there also is a positive relationship between the growth rate of human
capital and the interest rate. Recall that we get (2.21) by equating the growth rate

of total output production in (2.18) and the growth rate of consumption in (2.20). If

46

we have an increase in the interest rate, the consumption in the early period will be
relatively more expensive than the consumption in the later period. Consumers then
tend to reduce their consumption in the earlier period of time and raise it later. Thus
the growth rate in consumption increases with the increase in the interest rate. To
keep up with the increase in consumption growth, we must have an increase in total
output growth and, with the ﬁxed rate of population growth, this could be done only
by increasing human capital accumulation. Therefore, we have a positive relationship
between the interest rate and the human capital growth rate and this is represented
by the upward sloping CC curve in Figure 2.1 below. Another thing to note is that
an increase in the interest rate induces a larger increase in the human capital growth
rate with larger values of the elasticity of substitution. As the interest rate goes up,
we know that consumers would postpone their consumption. This would be much
easier with a higher value of the elasticity of substitution. Thus, an increase in the

value of 1/ d can be represented by a decrease in the slape of the CC curve.

r C

    

 

 

Figure 2.1: Equilibrium Human Capital Growth Rate

47

The equilibrium interest rate and the human capital growth rate will be deter-
mined at the intersection of the two curves. Even though Figure 2.1 above shows
that the CC curve cuts through the RH curve from below, it is drawn for a higher
value of d. A decrease in the value of 0 decreases the slope of the CC curve. Thus,
for a sufﬁciently high value of the elasticity of substitution (a low 0), we would have
the CC curve cutting through the HH curve from above. In fact, this would happen
when d < do 5(1-334' 31—p)/(1-17+ $)<1.

The equilibrium human capital growth rate can be algebraically derived by equat-

ing (2.19) and (2.21)

_ A-p+(d-1);-:7n
1+(a—1)(ﬁ+;33)

 

93

(2.22)

For this equilibrium human capital growth rate to be a meaningful interior solution,
we need to consider the following. From (2.1), we get 93/ A = 1 — 12. As 1 — v is
the fraction of human capital devoted to its accumulation, we need to assume that
0 < gg/A < 1. For the case of d = 1, the equilibrium growth rate for human capital is
simpliﬁed as g” = A—p, we need to assume that 0 < p < A. For other values of d > 0,
we need to assume that the population growth rate is in between two critical values
that are combinations of exogenously given parameter values. Those critical values
will differ depending on the values of the elasticity of substitution (d). In each case, we
need 0 < 9,, / A < 1 to be satisﬁed. Otherwise, the fraction of human capital devoted
to its accumulation will be either 0 or 1. If v = 0, all human capital is devoted to

its accumulation. Then there will be no human capital used in either manufacturing

48

or research activities. In this case, we will have no production, no research, and no
economic growth. On the other hand, if v = 1, there will be zero human capital
accumulation. As we note in (2.16), without human capital accumulation, we have
a zero growth rate in the number of new products. However, as we see in (2.18), we
still have economic growth as long as the population grows. In this case, we can no
longer call our model a R&D based endogenous growth model. In the analysis below,
we assume that the condition, 0 < v < 1, is always satisﬁed.

To get a steady state share of human capital employed in the research sector, we
need to go back to (2.13), which states the equality between the wage rate for human
capital and the value of marginal product of human capital in the research sector.
Multiplying both sides of (2.13) by H A and noting 6AfHA = A from (2.12), we can
rewrite (2.13) as w), - H A = PA - A. By substituting for PA from the no arbitrage

condition in (2.14) and rearranging, we can rewrite (2.13) as

 

- 7rA (2.23)

Substituting (2.9) into (2.8) and using (2.6) and (2.7), however, we can show that

«A: (a+ﬂ)(1—a-ﬂ)wh,

C!

HY (2.24)

Substituting (2.24) into (2.23) and rearranging it, we get

 

s 11,,_ AM ,(awxl-a-ﬂ)

= —. _ . 2.25
1-8 Hy r-PA/PA a ( )

49

where s is the steady state share of human capital employed in the research sector.6

From w), - H .4 = PA - A above, we can derive

E = 9..., + 9H - 9A (2'26)

From (2.6), we can also derive

g... = g, - g. (2.27)

Substituting (2.27) into (2.25), we get

15,, _
PA —gY 9.!

Substituting this together with (2.15), (2.16), (2.18), (2.20), and (2.22) into (2.25), if

we solve for s, we get

where, under the assumption 0 < 9,, / A < 1, we can show that

_ a (a —1)A + (a — 1)(1 - 413-331) — n)+(1- ¢)p
”'(awxl-a-m A-p+(a-1>s5n +1}

is always positive.

 

6Note that the total human capital available for the employment in either the research sector
or the manufacturing sector is vH, that is not devoted to human capital accumulation. Thus, 3
represents the fraction of human capital out of 121! .

50
2.4 Comparative Dynamics

In this section, we would like to study the long-run growth effects from changes in
the technology parameter of human capital accumulation (A) and from changes in
the population growth rate (n). Noting that the denominator in (2.22) can be either
positive or negative depending on the value of d, we can see that the growth effect of
A and n can also be either positive or negative.

Let’s consider the effect of the change in the technology parameter ﬁrst. An in-
crease in the value of A means that people become more productive in accumulating
human capital, as growth in human capital becomes faster even with the same pro-
portion of human capital (1 — v) is devoted to its accumulation. This could mean
many things. For example, the increase in A could come from an improvement in the
education environment such as better libraries, computer equipment, and internet
services.

We ﬁnd that such an improvement in the educational environment would increase
the long-run growth rate only when the consumption in two different periods of time
are not highly substitutable. In fact, only when d > do, would the increase in A
increase the long-run growth rate of the economy. Referring to (2.18), with a given
value of population growth, the long-run growth rate would increase when there is
an increase in the rate of human capital accumulation. We can see from (2.22)
that the growth rate of human capital is increasing in A when the denominator is
greater than zero. This requires that d > do. Otherwise, we would have decreases

in the human capital growth rate and the long-run economic growth rate as we have

51

a better environment for the education. Figure 2.2 below shows the effect of an

 
     

 

 

11 , i
H E
c E 5
9.“

Figure 2.2: Effect of Increase in A when d > do

increase in the value A on the human capital growth rate in the case of d > do. As
we mentioned earlier, the CC curve cuts through the HH curve from the below when
d > do. Recalling that the HR curve is from (2.19), we can see that the HH curve
will shift up as A increases. This increases the equilibrium growth rate for human
capital. Recalling that the equilibrium human capital growth rate in (2.22) is derived

by equating (2.19) and (2.21), if we rewrite (2.22) as

1 ﬂ _ 0
A+(1—¢—a+ﬂ)gﬂ+mn—U[(l-¢+ lgu-mnl'l’P‘l'n (2-28)

 

 

The left hand size represents the interest rate required for balanced human capital
growth and the right hand side represents the interest rate required for balanced

expenditure growth. As A increases, we have an increase in the interest rate. This

52

increase in the interest rate would mean an increase in the price of intermediate goods
used in the production of ﬁnal good, as we see from (2.9). This increase in the price
of intermediate goods reduces the ﬁnal good production as manufacturers of ﬁnal
good use a smaller amount of each intermediate good. This reduces the demand for
human capital in the ﬁnal good sector. The decrease in demand for intermediate
goods decreases operating proﬁts for each intermediate good producer and decreases
the patent price for each design. This decrease in the patent price is reﬂected in the
decrease in the value of the marginal product of human capital. This also reduces the
demand for human capital in the research sector. Thus the increase in the interest rate
decreases the demand for human capital from both manufacturers and researchers.
However, there also is a force that requires an increase in human capital. As the
right hand side of (2.28) represents, the increase in the interest rate induces higher
expenditure growth as consumers are trying to substitute their current consumption
for consumption in the future. The overall effect of the increase in the interest rate
on human capital depends on the relative size of these two opposing forces. As the
increase in the interest rate induces a smaller increases in human capital with a
lower value of consumer’s elasticity of substitution, we have an overall decrease in the
demand for human capital with higher values of d. In fact, if d > do, we will have
a decrease in demand for human capital. Unlike models in which no human capital
accumulation is presented,7 consumers in this model can decide the human capital

growth rate. As the overall demand for human capital decreases, more human capital

 

7For example, the growth rate in Romer’s model is determined by competing forces for a given
human capital by the research sector and the intermediate goods sector

53

is devoted to its accumulation. Thus, we have increases in the human capital growth
rate as shown in Figure 2.2 above.

On the other hand, if d < do, we will have a decrease in the equilibrium human
capital growth rate as the force that increases the demand for human capital outweighs
the force that decreases the demand for human capital. This can be shown as in Figure

2.3 below where the CC curve passes through the RH curve from above.

 
    

 

bcunnccocoooooouuuoc

 

Figure 2.3: Effect of Increase in A when d < do

Effects of the increase in the population growth rate on economic growth would
be somewhat different from that of the technology parameter A. If we differentiate

both sides of (2.18) with respect to n, we get

an =(l—d a+ﬂ)0n +a+ﬂ

 

(2.29)

As the part of the increased population directly goes to ﬁnal good production, the

increase in the population growth rate would directly increase economic growth The

54

second term on the right hand side of the equation above represents this. However,
the increase in the population growth rate also indirectly affects economic growth
through change in the human capital growth rate, as is represented by the ﬁrst term
on the right hand side of the equation above. As in the case of the change in A,
the effect of the change in the population growth rate on the human capital growth
rate could be either positive or negative depending on the value of the elasticity of
substitution. Let’s deal with this before we investigate the overall effect of the increase
in the population growth rate on the long-run economic growth.

As we see in the left hand side of (2.28), the increase in the population raises the
interest rate for balanced human capital growth. This shifts the HH curve upward
as in the case of the increase in the value of A. However, the analysis this time is
more complicated because we also have a shift in the CC curve as the right hand
side of (2.28) also contains n. The direction of the shift in the CC curve depends on
the value of 0. Figure 2.4 below is drawn for the case of d > 0’} E (a + [3)/a where
the CC curve shifts down as it grows.8 The downward shift in the CC curve can be
understood by noting that the CC curve is derived by equating (2.18) and (2.20).
From (2.18), we can see that the increase in the population growth rate raises the
output growth rate. The increase in 11 also raises the consumption growth rate, as we
note that d > 01 > 1 in (2.20).9 As d > 0'1, however, the increase in the consumption
growth rate is larger than the increase in the output growth rate.10 To balance these

two growth rates, we need to have a decrease in the interest rate and this is reﬂected

 

8As do < 1 < 0’1, we have the CC curve passes through the HR curve from below.
9For ago/0n =1-l/d > 0, we need that 0' >1.
1"From (2.18) and (2.20), agy/an < ago/0n requires that d > 01.

55

in the downward shift of the CC curve.

 
   

 

-.'\.-.-----

Figure 2.4: Effect of Increase in 12 when d > 01

The decrease in the interest rate means that current consumption becomes less
expensive relative to future consumption. Consumers then spend more and save less.
The growth rates of consumption and output decrease and thus the demand for human
capital decreases. This raises the human capital growth rate, as more human capital
is devoted to its accumulation. This is represented by the increase in 9,, up to the
point where the new C’C’ curve intersects with the RH curve in Figure 2.4. This
increase in the growth rate for human capital is further enhanced by the upward shift
in the HH curve. As the left hand side of (2.28) also contains 11, the increase in the
population growth rate calls for an increase in the interest rate. This is why the HH
curve shifts up. The explanation for the increase in the human capital growth rate
due to this shift is analogous to the case of increase in A, which we discussed earlier.

That is, noting that the left hand side of (2.28) is the equilibrium interest rate for

56

balanced human capital growth, the increase in the population growth rate raises the
interest rate and this decreases the demand for human capital from both the ﬁnal
good sector and the research sector. As the demand for human capital is decreased,

we have an increase in the human capital growth rate.

  
   

 

 

Figure 2.5: Effect of Increase in n when 1 < d < 0'1

If 0 < 0'1, we will have the CC curve shifting up instead of shifting down.ll
Continuing on the case of d > do so that the CC curve passes through the HR curve
from the below, an upward shift in the CC curve causes a decrease in the human
capital accumulation rate, with the reasoning opposite to the one given in the case
of d > 01 above. However, as the HH curve also shifts up, we have a force requiring
an increasing the growth rate of human capital. The overall effect of an increase in
the population growth rate on the human capital growth rate would depend on the

relative magnitude of these two opposing forces. It turns out to be the that the force

 

llWhen d = 01, the CC curve does not shift. However, as the RH curve still shifts up, we will
have an increase in the human capital growth rate.

57

that increases the human capital accumulation rate outweighs the other force, when
d > 1 as the RH curve shifts more than the CC curve. Thus, in this case, we will have
an increase in the human capital growth rate as the population growth rate increases
as in Figure 2.5 above. If do < d < 1,12 on the other hand, we will have a decrease in
the human capital growth rate since the CC curve shifts more than the HR curve.
Finally, for the case d < do, we have the CC curve pass through the HH curve
from the above. The increase in the population growth rate shifts both the CC and
the HH curves upward. Thus the shifts of the two curves affect the human capital in
opposite directions. However, as we have a < 1, the CC curve shift more than the
HH curve. Thus, in this case, the increase in the population growth rate raises the

human capital growth rate as shown in Figure 2.6 below.

   

O

 

Figure 2.6: Effect of Increase in 11 when d < do

Now we can see that the increase in the population growth rate increases the

 

12When 0' = 1, we have 9,, = A - p. Thus, the human capital growth rate does not depend on the
population growth rate.

58
growth rate of human capital accumulation for all values of 0, except for the inter-
mediate range do < d S 1.

Aside from this range, the increase in the population growth rate increases the
human capital growth rate and this helps the long-run economic growth in addition to
the direct effect through the increase in the employment of human capital in the ﬁnal
good sector as indicated by the second term on the right hand side of (2.29). In the
case of do < d S 1, the direct employment effects and indirect effects through human
capital accumulation work in opposite directions. By differentiating the equilibrium
human capital growth rate in (2.22) with respect to n and substituting the result
into (2.29), we can show that the direct employment effect is larger than the indirect
effect when 02 < d < 1 where 0'2 E (1‘5? + 324%)“ﬁ + 3&5). Thus,’we can say that
the increase in the population growth rate increases the economy’s long-run growth
rate for all ranges of 0‘, except for do < d S 02.

In this section, we show that growth effects of an improvement in the educational
environment and an increase in the population growth rate critically depends on
the value of the elasticity of substitution. However, Deaton [1992] reports various
measures of its value between 0.265 and 0.736.13 According to these ﬁgures, we have
d in the range of 1.36 to 3.77 and we can say that an improvement in the educational
environment and an increase in p0pulation growth rate increase the long-run economic
growth rate. However, it is an empirical matter and, as Deaton pointed out, we don’t
have good estimates of actual intertemporal consumption responses. Thus, the result

in this section remains as a possibility.

 

”See Understanding Consumption, 1992, p.73.

59
2.5 Conclusion

Jones [1995b] uses the term of semi-endogenous growth in describing his model, since
R&D activities are done by proﬁt seeking ﬁrms while the long-run economic growth
rate is determined by exogenously given parameter values like the population growth
rate. Long-run economic growth in the model presented here is also determined partly
by the population growth rate. However, long-run economic growth in this model is
also affected by the endogenously determined human capital growth rate. Thus, the
model presented here can be called a trueendogenous growth model, in the sense
that the long-run growth rate is determined, at least by, human capital investment
decisions. This feature is also presented in Segerstrom [1995]. However, innovations in
his model are different as innovators are trying to improve the quality of the existing
products.

Consumers’ elasticity of substitution in this model play an important role in de-
termining long-run economic growth. An increase in the technology parameter for
human capital accumulation (A) increases the interest rate, and this increase in the
interest rate raises the consumption growth rate, as consumers substitute current
spending for that at a later point in time. The increase in the consumers’ expen-
diture growth rate requires more human capital to be devoted to the production of
ﬁnal good. On the other hand, increases in the interest rate calls for less human
capital as the demand for that from both the manufacturing of the ﬁnal good and
the research sector decreases. Thus, there are two opposing forces for the demand for

human capital and the dominance of one force over the other depends on the value of

60

the elasticity of substitution. As the substitution of current consumption for future
consumption is easier with higher values of the elasticity of substitution, the force
that calls for more human capital in the ﬁnal good sector is larger when we have a
higher value of 0. Unlike models in which no human capital accumulation is present,
consumers in this model can decide the human capital growth rate. As forces that
demand more human capital in the production side“ outweighs that demand less, it
is natural that consumers devote a larger proportion of human capital to production
activities and smaller proportion to human capital accumulation. This decreases the
growth rate for human capital and, consequently, the long-run economic growth rate.
This ﬁnding has an interesting implication that better environment for the education
does not necessarily lead to faster growth in the economy.

Effects of increases in the population growth rate on the equilibrium human cap-
ital growth rate also depend on the value of the elasticity of substitution. Unlike
changes in A, in which the initial effect on the interest is only on the human capital
accumulation, changes in the population growth rate affect the interest rate required
for balanced growth in both human capital and expenditure. Although the analysis
in this case becomes little bit more complicated due to the changes in the interest
rate on both sides, it is basically the same as the above case in the sense that the
relative dominance of two opposing forces depends on the value of the elasticity of
substitution. It turns out to be the case that, for most of values of the elasticity of
substitution, the force that calls for more human capital growth outweighs the other

force except for some intermediate range of the elasticity of substitution. Therefore,

 

“The ﬁnal good and the research sectors.

aside from this intermediate range, increases in the population growth rate increase
the human capital growth rate.

As the increased population is absorbed into productions of the ﬁnal good, changes
in the population growth rate also directly affects the long-run economic growth rate.
Except for the intermediate range of the elasticity of substitution mentioned above,
therefore, increases in the population growth rate increase the long-run economic
growth rate through both the direct and the indirect effects. In the intermediate range
of the elasticity of substitution, the direct effect increases the long-run growth rate
and the indirect effect decreases the long-run growth rate. The direct eﬂ'ect dominates
the indirect effect for larger values of d in this range. Thus, we can also say that the
increase in the population growth rate increases the long-run economic growth rate
for most of values of the elasticity of substitution, except for some intermediate range.

As mentioned above, the model presented here improves Jones’ model in the
sense that the long-run economic growth rate is determined, at least partly, by the
endogenously determined human capital growth rate. However, the model presented
here is still like Jones’ model in the sense that no policy parameters like tax and
subsidy rates appear in the determination of the equilibrium growth rate. However,
the ﬁndings in this model suggest the possibility of developing endogenous growth
models without scale effects but with policy parameters affecting long-run economic
growth. Further exploration of models with decreasing returns to scale in R&D
is needed to see whether it is actually possible to develop models in which policy

parameters affect the long-run growth rate.

61

Chapter 3

Factor Endowments and Economic
Growth in a Model With

Technology 'Ii'ade

3.1 Introduction

Since the publication of a seminal paper by Vernon [1966], there have been many
attempts to incorporate his idea of an international product cycle into a formal dy-
namic model. Krugman [1979] considers an economy with two regions, innovating
North and imitating South, and presented a model in which a constant fraction of
goods are produced in each region. However, innovation and imitation rates in his
model are given exogenously. Dollar [1986] presents a model that allows endogenous
technological diffusion, while the innovation remains exogenous. With the develop-

ment of new growth theory in the 19803, that emphasizes the increasing returns to

62

63

scale and imperfect competition, Segerstrom et al. [1990] present a model that allows
endogenous innovation in a model of product cycle. While imitation is exogenous in
their model, in the sense that patent length is exogenously given, Grossman and Help-
man [1991] present models that allow both endogenous innovation and endogenous
imitation.

The model presented here involves an issue of how changes in labor endowments
affect the long-run growth rate of the economy. In a horizontal product differentiation
framework, Grossman and Helpman show that only the expansion of the southern
effective labor force has a positive growth effect.l Introducing two types of labor
(Human Capital and Unskilled Labor) in the Grossman and Helpman’s wide-gap
case, Lai [1995] shows that the expansion of southern human capital has a positive
growth effect, whereas the expansion of southern unskilled labor has a negative growth
effect. However, changes in either labor supply in the North continue to have no
growth effect.

As all the models mentioned above consider imitation as the only vehicle of tech-
nological diffusion between the North and the South, they tend to ignore the rapid
growth of technology trade between countries in recent years. For example, US.
earnings from royalties and licensing fees abroad has grown from $5.1 billion to $15.3
billion during the period between 1982 and 1990.2 By making the assumption that
imitation is costlier than purchasing technology, Liu [1994] shows that there exists a

long-run steady state equilibrium with technology trade. In his model, expansion of

 

1This is a result for the wide-gap case in their model. In the narrow-gap case, expansion of
effective labor in both regions has a positive growth effect.
25cc Survey of Current Business, June 1991, p.45.

64

the labor forces in both regions has a positive growth effect.

Introducing two types of labor in Liu’s model, as in Lai, this paper shows that
the expansion of diﬂ'erent types of labor in each region have different growth effects.
Increases in human capital in each region has a positive growth effect, whereas in-
creases in northern unskilled labor negatively affect long-run growth. Increases in
southern unskilled labor could be either positive or negative. Results in this paper
are new, compared to Lai, as changes in northern labor forces now have long-run
growth effects. Unlike Liu’s model, where only one type of labor is considered and
increases in labor endowments in either region always have a positive growth effect,
the model presented here has results more in depth.

In Section 2 below, the structure of the model is described. The solution of the
model for a steady state equilibrium with positive growth rates is developed in Section
3. Then in Section 4, comparative dynamic analysis is done for each type of labor in
each region. Finally, in Section 5, we summarize implications of the ﬁndings of this

paper.

3.2 The Model

At time t, nN(t) products are produced in the North and n5(t) products are produced
in the South. Goods are freely traded across the border. All the n(t) = nN(t) + ns(t)
products are developed in the North. That is, only the North is capable of designing
new products which are horizontally differentiated from the existing products. The

diffusion of technology from the North to the South is as speciﬁed in Liu [1994]. That

65

is, some proportion of the products developed in the North immediately ﬁnds produc-
ers in the South. Even though the South is capable of imitating northern developed
products, because of high imitation costs, it is more proﬁtable for southern manu-
facturers to buy technology directly from northern innovators. Thus, imitation never
occurs. Also as in many R&D based growth models3, if two or more producers are
producing the same product, each ﬁrm earns zero proﬁt due to Bertrand competition,
and cannot recover initial development costs. Thus, a single ﬁrm, located either in

the North or in the South, produces each product in equilibrium.

3.2.1 Consumers
A representative consumer in the world (in the North and the South), who lives
forever, maximizes the intertemporal utility
U(t) =-_- / e“’("’)logu(r)dr (3.1)
1

subject to intertemporal budget constraints

[00 e’[R(')"R(’)]E(r)dr S [00 e’[R(’)‘R(’)lY(r)dT + A(T) (3.2)
t

t

where p > 0 is the consumer’s subjective discount rate, R(t) is the cumulative in-
terest rate, Y(t) is the consumer’s income, and A(t) is the asset value. Also, the

instantaneous utility, n(t), and the instantaneous expenditure level, E(t), are given,

 

3For example, Grossman and Helpman [1991]

66

respectively, as

um a If") scorer” (3.3)
and

em = [mponowj (3.4)

where 0 < a < 1 and :r( j) and p( j )4 are, respectively, the consumption and the price
of product j.

As the intertemporal utility function in (3.1) is time separable, we can solve the
consumer’s problem in two stages. That is, for the expenditure level given in (3.4),
we can ﬁrst solve for the maximization of the instantaneous utility in (3.3). In doing
so, we get the instantaneous demand for product j as

. 100')“
:r j = . _ . 3.5
( ’ 16‘ pm! edr ( )
where c E 1/ (1 — a) is the elasticity of substitution between goods and E is the

aggregate world expenditure.5 Substituting (3.5) into (3.3), (3.3) into (3.1), and

maximizing (3.1) subject to (3.2), we get the consumer’s optimal expenditure path

as

 

‘As goods are freely traded, the price of a good is the same for consumers and producers.

5Consumers are homothetic, the representative consumer’s behavior is proportional to the aggre-
gate behavior of consumers in the world. Thus, we can interpret E and 2:( j) in (3.5), respectively,
as the aggregate world expenditure and the aggregate demand for good j.

67

where, throughout this paper, dots above variables represent the time derivatives of
those variables. As in Lai [1995], imposing normalization E (t) = n(t) for all t, we

ﬁnd that

R=p+g 66)

where g 5 12/71.

3.2.2 Innovation

Innovation in this model is in terms of ﬁnal goods and it occurs only in the North. The
innovation process is described as in Romer [1990]. That is, based on public knowledge
I: = n at the time, innovators in the North develop designs for new products by using

human capital according to

R
HN-n

an

 

{1:

(3.7)

where H S is northern human capital used in the research sector and a o is the pro-
ductivity parameter for the innovation. With this speciﬁcation, denoting wﬁ to be
the wage rate for human capital in the North, the unit cost of a new product design
will be 112a o /n. Free entry into the research sector ensures that the price for a new
product design equals the unit cost. Thus, denoting P3 as the price of the blueprint

for a new product, we have

 

68
3.2.3 Northern Production

The setup cost for a northern manufacturer is the purchasing cost of a product design

from an innovator. Once paid P3 for a product design, a northern manufacturer tries

to maximizes the discounted present value of lifetime operating proﬁts

00
t

H~(t) = / e‘lR(’)'R(’)lnN(r)dr (3.9)

where the northern instantaneous operating proﬁts, 1r "(1'), are

"111(7) = p~(T)x~(T) - C~(T)x~(‘r) (3-10)

in which 1),, and c" are, respectively, the price and the unit production cost of a
northern product. Given the demand for a northern product, .1: N, from (3.5), the

mark-up price for a northern product can be calculated as

 

_ 6.1!
PN - a (3.11)
Substituting (3.11) into (3.10), we get
1 _
7r” - achN (3.12)

As in Lai [1995], manufacturers in the North use both human capital and unskilled

where 0 -
of human
productiv
this prod;

Cothm.

3.2.4

 

The same
HOWQVQI?
Chasing h
PYOducin
from the
latioa 0

aL/ns L

69

labor. That is, the production function for a good i is given as

ma) = AlﬂhU)” + (1 — ﬂ)l(i)’1”’ (3.13)

where 0 < H < 1, —1 _<_ 7 S 0,6 and h(i) and I(i), respectively, are quantities
of human capital and unskilled labor devoted to the production of good i. A is a
productivity parameter. Note that the elasticity of substitution between inputs in

this production function, 5 E 1 / (1 —7), takes the highest value of 1, when 7 = 0 (the

Cobb-Douglas case).

3.2.4 Southern Production

The same production technology in (3.13) is also applied to southern manufacturers.
However, southern manufacturers have to go through an additional process after pur-
chasing blueprints of product designs from northern innovators and before actually
producing the products. This process involves a learning cost such as those coming
from the training of local labor, education for managerial level employees, or trans-
lation of blueprints into the local language. As in Liu [1994], this process requires
a 1. / n 3 units of human capital,’ where a 1. is the productivity parameter for the learn-

ing activity. The set up cost for a southern manufacturer will then be the sum of

 

°In Lai, 7 S 0 is assumed to ensure that both types of labor are used in manufacturing. In this
model, 7 S 0 is continued to be assumed. However, algebraic complication arising in this model
forces us to impose an additional assumption 7 2 —1. As a result, the production function in (3.13)
cannot include the Leontief case (7 = -oo). However, our assumed range of 7 continues to include
the Cobb-Douglas case (7 = 0).

7In Liu, labor is used for this process. However, as labor in his model is used for the innovation
in the North, human capital in this model is analogous to labor in his model.

70

the purchasing cost of a blueprint, P3, and the learning cost, wiaL/ns, in which
we denote w; as the wage rate for human capital in the South. Also as in Liu, for
North-South technology trade to make sense, we need to assume that imitation by
the South costs more than the setup cost. That is, if the South can imitate northern

developed product according to vi, = H g - n 5 / a ,, it should be assumed that

wia, wiao

 

 

>PB+

n5 "5

where a, denotes the productivity parameter for southern imitation.

Once incurring the setup cost, a southern manufacturer tries to maximize the

discounted value of lifetime operating proﬁts

IIs(t) = [00 e’[R(’)'R(’)l1rs(r)dr (3,14)
1
where the southern instantaneous operating proﬁts, 71'5(r) is
«5(7) = p3(r):rs(r) — c3(r):rs(‘r) (3.15)

in which p5 and Co are, respectively, the unit price and the unit production cost of a
southern product. Given the demand for a southern product in (3.5). the mark-up

price for a southern product can be calculated as

P5 = Z (3.16)

71

Substituting (3.16) into (3.15), we get

 

csxs (3.17)

3.2.5 No Arbitrage Condition

As there are many potential manufacturers for a product design in the North, inno-
vators in the North can charge the blueprint price that equals the discounted value

of a manufacturer’s lifetime Operating proﬁts. Thus, we have

H~(t) = P190) (3-18)

Substituting (3.8) and (3.9) into (3.18) and doing logarithmic differentiation with

respect to time, we get the northern no arbitrage condition as
- r'z
——=R-—+; (3.19)

This condition indicates that the instantaneous operating proﬁts for a northern man-
ufacturer must cover the interest rate and the instantaneous net capital loss.
As for the southern manufacturer, the blueprint price must equal the discounted

value of a manufacturer’s operating proﬁts minus the learning cost. That is,

wiab

 

use) = P3(t) + (3.20)

ns

72
Substituting (3.8) and (3.14) into (3.20) and doing logarithmic differentiation to (3.20)

with respect to time, we get the southern no arbitrage condition as

 

 

'N 's
N w s w s s
w a n w a n N a 5 2:.
n5 _ . _ H a 555+ 11 1 3% wﬂaonsn-i-wﬂabnns (3 21)
I13 wgaons + wflaon wgaons + wgabn

As in the no arbitrage condition for the North, the condition in (3.21) indicates that
the instantaneous operating proﬁts must cover the instantaneous operating proﬁts and
net instantaneous capital loss. Otherwise, southern manufacturers cannot recover the

initial setup cost and will consequently earn negative proﬁts.

3.2.6 Labor Market

Finally, to close the model, we need to describe the labor market. Assuming full

employment for both types of labor in each region, we can write the northern human

capital market clearing condition as
7’ P
as; + HN = HN (3.22)

where the ﬁrst term in the left hand side is the human capital employed in the research
sector and derived from the innovation function (3.7). H S and HN, respectively,
represent the human capital employed in the northern production of manufacturing
goods and the total human capital endowments in the North. Denoting L N as the total
endowments for unskilled labor, the market clearing condition is simply Li, = LN,

since unskilled labor is only used in the production sector. Similarly, for the southern

73

region, we have the market clearing condition for human capital as
aL_+HS = HS (3.23)

where the ﬁrst term in the left hand side is the human capital devoted to the learning
process, H g is the human capital devoted to the actual production process, and H5
is the total human capital endowment for the South. Also, denoting L5 as the total

unskilled labor for the South, we can write the market clearing condition for southern

unskilled labor as LP = L 5.

3.3 Steady State Equilibrium

In this section, we solve for a steady state equilibrium where the growth rate of the
number of products, 9 E ii / n, and the fraction of goods produced in each region,
0‘ E nN/ns, are constant over time. Recalling our normalization, E / n = 1, then, we
have 9 = 12/71 = rig/n5 = wﬁ/wfj = wﬁ/wfr Imposing these conditions in (3.19) and

(3.21), we have no arbitrage conditions for the North and the South, respectively, as

7f
If); — p + g (3.24)
and

s .—
Us - p + g (3.25)

74

As the expenditure level and the number of products grow at the same rate, with
the normalization, E /n = 1, we have capital loss terms cancel out in (3.19) and
in (3.21). Thus, the no arbitrage conditions in (3.24) and (3.25) simply imply that
the instantaneous operating proﬁts in each region need to cover the interest rate.8
Note that the southern no arbitrage condition in (3.25) is the same as in Lai [1995].
However, the northern no arbitrage condition is different. In Lai’s model, northern
manufacturers face the risk of their products being imitated by southern imitators
and losing the market to southern manufacturers with cost advantage. However, in
this model, such imitations are not possible and we don’t have the term, 11, that
reﬂects the hazard rate.9

Substituting (3.12) and (3.11) along with (3.8) in (3.24), we get

-%— = n+9 (326)

Denoting 45,, as the factor cost share of human capital in the northern manufacturing

sector, we get

NHP
chN = w” N (3.27)
nN¢N

 

Noting symmetry among northern manufacturers and using (3.22), from (3.13), we

also get

MHN " “0.9)"

’” = 12(HN - apgr + (1 - 19L},

(3.28)

 

I’wmi E/n = 1, R = p + 9. See (3.6).
9See equation (5) in Lai.

5111351112;

l5?

:3
U)

Ihn eq
markets

{3533i 11

and this

1

Appe

>-

A‘.

which ir
”Til am
equilibri
HOldlng
capital 1
This [DC]
equals 1]

increase

and (he I

ll.
H15 mc

75

Substituting (3.28) into (3.27), (3.27) into (3.26), and then rearranging we get

n_,. _ a = (1 - axHN - any)! -~16(H~ — apgr +(1— (M N]
no 000,300 + 9)"(1_ OXHN - aog)’ IlﬂUIN "' €109)” +( 1" ﬁlLN l

(3.29)

This equation satisﬁes the equilibrium condition in the goods, capital, and labor

markets in the North. Note that, if we set [3 = 1, (3.29) is reduced to the equation
(2-23) in Liu

= (1 - axHN/ap — g)
cm + g - <1 — axHN/a.)

and this indicates that his model is special case of the model presented here.10 In

Appendix, it is shown that total differentiation of (3.29) gives

dd
dgN

which indicates that there is a negative relationship between the growth rate (9 -
Mn) and the fraction of goods produced in the North (0 = nN/ns) along the northern
equilibrium condition (3.29). The negative relationship is understood as follows.
Holding everything else constant. an increase in 9 increases the demand for human
capital in the North. This increases the wage rate for human capital in the North.
This increases both the unit cost of product development and, as the blue print price
equals the unit cost of product development, the setup cost for manufacturers. The
increase in the wage rate for human capital also increases the manufacturing cost

and the price of goods in the North. Thus, the per product sale and proﬁt decrease.

 

u’His model considers the economy with only one type of labor.

Both 1
new 11
goods.

the 111

(3.2971.

Sub

Condl' 11

r?

DenOIin

76

Both the increase in the setup cost and the decrease in the proﬁt discourage entry of
new manufacturers in the North and decrease the fraction of northern manufactured
goods. Thus, we have a negative relationship between 9 and 0. Also, by considering

the limit case of d —> oo in (3.29), we can see that g asymptotes to the value satisfying

(1— mg} - g = ap — (1 - Ct)(1 E'BXyﬁ' " 9)"7(ﬁ_:)7

 

The N N curve in the Figure 3.1 below represents the northern equilibrium condition

(3.29), based on our ﬁndings above.

SS

    

 

 

8-11-

14

Figure 3.1: Steady State Equilibrium

Substituting (3.17) and (3.20) along with (3.8) into the southern no arbitrage

condition (3.25), we get

 

wNa wsa =p+g (330)
494.1%!-

Denoting ¢s to be the factor cost share of human capital in the southern manufac-

77

turing sector, we get
P
wZHs

ns¢s

 

651:5 =

wan

Noting symmetry among southern manufacturers and using (3.23), from (3.13), we

also get

ﬂ(Hs — (11.9)!y

4" = ﬂ(Hs - aLgr +(1- m2

 

(3.32)

Substituting (3.31) and (3.32) into (3.30), we get

1-.. wgyg n(Hs—aigmu-ML}

a n5 mHs-GLQP

wNﬂap + zusﬂaL
1: n5

=p+g

Substituting (3.23) for H 5 in the above and rearranging terms, we get

l—a

 

 

(H5 - mgr-"mm. - any)” + (1 - [3)in = «.mp + g)[(‘”i/wi)(“°/“‘) + 11

a n/ns

(3.33)
The wage rate for human capital in each region equals the value of the marginal
product of human capital in each manufacturing sector. Thus, noting symmetry

among manufacturers in each region in (3.13), we get

LN

N- ”-93. _ __
wH—VMP -05Alﬂ+(1 ﬂXHN-apg

)wll-w/w

and

Ls

5— 5 _C_5 I _ —_
wH—VMP —aﬁA[d+(1 BXHs-aLg

)‘vll-‘v/‘v

Dividi:

(dug

(sage

Subsut

 

 

 

 

Subsﬁt

conditi.

78

Dividing the former by the later, we get

my _ a 3+(1-ﬁ)(H—:iﬁ)7

1-7/‘7
w; ‘ cs 3+ (1 — xix—LL)” (3'34)

Hs-aLy

Using (3.5), (3.11), and (3.16), we get

CN3N

 

_ C_N)l-¢
053's Cs

Using chN = wfjHﬁ/nNQSN and c523 = wZHg/nsqﬁs, we can rewrite the above as

 

c” ngﬁ/n~¢,v 1/1_¢
_ = 3035
CS (wﬁHg/nscbs) ( )

Substituting (3.35) into (3.34) and using (3.22), (3.23), (3.28), and (3.32), we get

 

w: _ 72—” l-a HS—abg 1-0, 13+(1—BXH—Sii—L-yY 0(3_?-;3_1%1)
E— 715) (HN-aog) [3+(1—ﬂ)(__u_HNL_aD:)w] (3-36)

Substituting (3.36) into (3.33) and noting a = n N / n 5, we get the southern equilibrium

condition as

l-a L5

(HS - angllﬂ + (1 - ﬂ)(

 

HS-aLg) I:

L
01-a( Hs-a” )l-a[ﬁ+(l-ﬁ)(H—SE§E)7 10(12—°-L?)(22)
HN-aoy a+(1_g)(.”N__.¥.F5)w aL

0L3(P+9){ 1+0,

+ 1} (3.37)

5.]

'9

as )0;

Theﬁ

needec

 

79

This equation satisﬁes equilibrium conditions in the goods, capital, and labor markets

in the South. Setting ﬂ = 1, we can also show that (3.37) is reduced to equation (2-

24’) in Liu.

a H —a l—a
03110) + g)(0-5—”HN-.D,)

(1-0)':'f-ap-g

-1

As shown in Appendix, total differentiation of both sides of (3.37) gives a positive

relationship between the growth rate and the fraction of goods produced in the North

along the southern equilibrium condition,

dd

0
dg >

S

as long as a > a/(l — a), Ls/aL > LN/aD, Hs/aL > HN/ap, and —1 S 7 S O.11
The ﬁrst two conditions are also assumed in Liu, while the last two conditions are
needed to accommodate newly introduced structure in this model. The ﬁrst condition
requires that, for a given value of a, and thus, for a given value of the elasticity of
substitution between goods, 6 = 1/(1 — a), a relatively larger fraction of existing
goods are produced in the innovating North. The second and the third conditions
require relatively large effective labor forces in the South. These conditions can
be guaranteed by assuming that, for given labor endowments in each region, the
southern learning activity for already developed product design is much easier than
the northern innovating activity of designing new products (aL < a D). As explained

in Section 2.3, the last condition restricts us in the intermediate range of the elasticity

 

11Outside the range of these assumptions, the sign of da/dg is ambiguous.

is. C61

huma:

manuf

have a

relatio

WC Cdi'

 

The S
on our
produ.

Curve:

3.4

In th

Into 7

80

of substitution between inputs in manufacturing. The explanation for this positive
relationship between 9 and a is similar to that in the northern equilibrium case. That
is, ceteris paribus, increases in 9 make the demand and the wage rate for northern
human capital increase. This results in an increase in the setup cost for southern
manufacturers and decreases the entry of new manufacturers in the South. Thus, we
have a decrease in the fraction of goods produce in the South (0 T) and a positive
relationship between 9 and a. By considering the limiting case of a —+ co in (3.37),

we can see that g asymptotes to the value satisfying

(1— 3%:- —g = up -(1— a)(1 7ng -g)‘v(§f—)~ (3.38)

 

The SS curve in Figure 3.1 represents the southern equilibrium condition, based
on our ﬁndings above. The equilibrium growth rate and the fraction of products

produced in each region are determined at the intersection of the SS and the N N

curves as in Figure 3.1.

3.4 Comparative Dynamics

In this section, we will consider the effect of increases in human capital and unskilled
labor in each region on the growth rate of the number of products. Substituting (3.5)

into (3.3) and differentiating it with respect to time, we can show that

_l—an

 

an

Silt?-

Thus
of 2135

gott'il

3.4.1

As sho1

under<
curve 5
in norti
North.
This re
ufactur»
means :
rate in

NOrth.

 

(V :) a:

 

81

Thus, the growth rate in the instantaneous utility is proportional to the growth rate
of the number of products, and the results in this section can be considered as the

growth effect of consumers’ welfare in the world.

3.4.1 Increases in Northern Human Capital (H N T)

As shown in Appendix, holding g and LN constant, total differentiation of (3.29) gives

do > 0

under our assumption that 0 < [3 < l and —1 S 7 S 0. This means that the NN
curve shifts up12 as H N increases. Holding g constant, for given n and n", increases
in northern human capital increase the human capital available for production in the
North. This reduces the cost of production and the price of goods in the North.
This results in increases in per product sales and operating profits for northern man-
ufacturers. Referring back to the no arbitrage condition of the North in (3.24), this
means that the per manufacturer proﬁt rate in the left hand side exceeds the interest
rate in the right hand side. This encourages the entry of new manufactures in the
North. Thus, we have increases in both the fraction of goods produced in the North
(0' T) and the growth rate in the number of products( g T). This is represented by the
upward shift of the N N curve as in Figure 3.2 below.

Increases in northern human capital also affects the southern equilibrium condi-

 

12This is equivalent to saying that the N N curve shifts to the right.

non.l:

under
up as
aﬁects
Hohhn
theun
theSO
alone
SOUIhe
both t

“Umbe

82

SS '
SS

 

 

Figure 3.2: Growth Effect of HN T

tion. In Appendix, we show that total differentiation of (3.37) gives

.22. < 0
dHNLs

(under the assumption that a > (1 - a)/a. This means that the SS curve also shifts
up as HN increases, as in Figure 3.2 below. Increases in northern human capital
affects the southern equilibrium condition through the decrease in the blueprint price.
Holding g constant, for given n and n 5, increases in northern human capital lowers
the unit cost, and thus the blueprint price, of a product design. As manufacturers in
the South also have to buy northern product designs, the lower blueprint price means
a lower setup cost for southern manufacturers. This encourages the entry of new
southern manufacturers. Thus, increases in northern human capital tend to increase

both the fraction of goods produced in the South (0 l) and the growth rate of the

number of products (9 T).

83

As increases in northern human capital shift both the N N and the SS curves
upward in Figure 3.2 above, their growth enhancing effects reinforce each other and
we have an unambiguous increase in the growth rate.

The result here is new compared to that in Lai [1995]. In his model, the setup cost
for southern manufacturers is independent of the cost of product development in the
North. That is, the denominator of the left hand side of (3.30) changes to wﬁa , / n 5.13
Then, for a given 9 and n from the North, the number of products manufactured in
the South, ns, and thus, the fraction of goods produced in each region, 0' = n ~/ n 5, are
solely determined by the southern composition of labor.” Thus, the SS curve in Lai’s
model becomes horizontal. Furthermore, as the set up cost for southern manufacturers
is independent of the wage rate for human capital in the North, changes in northern
human capital, and the resulting changes in the wage rate, do not shift the SS curve.
Without shifts in the horizontal S'S' curve, shifts in the N N curve alone cannot lead
to changes in 9.

With the collapse of the old Soviet Union, we see that lots of skilled labor from
eastern European countries migrate to the western world and work for the market
economies. We can consider this as an increase in human capital in the innovating
North. Unlike Lai’s model, the model presented here can project a positive growth

effect of such an increase in northern human capital.

 

13See equation (7) in Lai.
“w?“ mi, and as are determined by the southern composition of labor.

84

3.4.2 Increases in Northern Unskilled Labor (LN T)

In Appendix, holding g and H N constant, we show that the total differentiation of

do
—-‘ < 0
dLN N _-

which means that the N N curve shifts down as LN increases.15 As northern un-

(3.29) gives

skilled labor increases, the manufacturing cost and the price of northern products
decrease. This increases the per product proﬁt rate and encourages entry of manu-
facturers in the North. This tends to increase both the fraction of goods (0 T) and
product development (9 T) in the North. However. at the same time, increases in
northern unskilled labor increase the marginal product of human capital in northern
manufacturing. This increases the wage rate for human capital in the North. This
tends to increase the manufacturing cost and offsets the above effect of encouraging
entry of new manufacturers. Furthermore, since the research sector only uses human
capital, the unit cost of developing new products increases and the price for a new
blue print goes up. This increases the setup cost and further discourages the entry
of new manufacturers in the North. As the second effect, through the increase in the
marginal product of human capital, outweighs the first effect, increases in northern
unskilled labor decrease both the fraction of goods produced in the North (0 l) and

the rate of product development (9 I). This is represented by the downward shift in

the NN curve in Figure 3.3.

 

15When 7 = 0, increases in LN do not affect the northern equilibrium condition. The same is also
true in Lai.

85

SS
33 '

 

 

Figure 3.3: Growth Effect of LN T

Total differentiation of (3.37), which is also given in Appendix, gives

do
__ > 0
dLNLs _

which means that the SS curve shifts down as LN increases.16 As mentioned above,
increases in northern unskilled labor increase the marginal product of human capital
and the blue print price of a new product in the North. As southern manufactures in
this model also need to buy blue prints from the North, this also increases the setup
cost in the South. Then, the entry of new manufacturers in the South is discouraged.
Thus, the fraction of goods produced in the South decreases and the rate of product
development also decreases. This is represented by the downward shift in the SS curve

in Figure 3.3. As both the N N and SS curves shift down, we have an unambiguous

 

mAs in the northern equilibrium condition, when 7 = 0, the increases in LN do not affect the
southern equilibrium condition.

86

decrease in the growth rate.

This result is also new compared to Lai. As explained in the case of increasing
northern human capital, the set up cost for southern manufacturers is independent of
the cost of product development in the North. Thus, increases in northern unskilled
labor and the resulting changes in the cost of product development in the North do not
affect the southern equilibrium condition. Increases in northern unskilled labor do not
shift the SS curve in Lai’s model and, since the SS curve in his model is horizontal,
the shift in the N N curve alone cannot result in changes in the equilibrium growth
rate. The result here are also different from Liu [1994]. In his model, increases in
northern labor have a positive growth effect. However, as he considers only one type
of labor, the effect through changes in the relative marginal product between two
different inputs cannot be addressed. Especially, as one type of labor is used in both
the manufacturing and the research sectors, increases in northern labor in his model
are equivalent to increases in northern human capital, which are used in both the
research and the manufacturing sector in this model.

The ﬁnding in this section has an implication related to the productivity slow
down in the 19808. It is reported that there has been a sharp increase in the fraction
of high school graduates relative to college graduates in the US. labor force in the
19803." Although we do not have population growth in this model, if we consider
this increase in the fraction of high school graduates as an increase in the unskilled

labor force in the US, which obviously is a part of the innovating North, we can

 

1“'Between 1979 and 1987, the number of 25-34 year old male high school graduates in the US.

labor force grew by 40 percent, compare to an only 32 percent increase in the number of 25-34 year
old male college graduates. See Levy and Murnane [1992].

87
expect a decrease in the world growth rate. This implication cannot be addressed in

either Lai or Liu.

3.4.3 Increases in Southern Human Capital (H5 T)

The northern equilibrium condition in (3.29) does not contain the southern human
capital term, (H 5), so increases in southern human capital do not affect the northern
equilibrium condition. Thus, when there is a change in southern human capital, the

N N curve stays the same. Holding g, H N, LN, and L5 constant, total differentiation

of (3.37), which is shown in Appendix, gives

do
mLS < 0

which means that the SS curve shifts up as H 5 increases. As southern human capital

 

 

g
38'
SS
...... g.‘
:i m
f '1 a
Q
1-c.

Figure 3.4: Growth Effect of H3 T

increases, the wage rate for human capital and the cost of learning for southern

manufact i
and increa
we have a:
In Liu’s rr
activities.
human ca;
qualitative

capital in ',

3.4.4 I.

Like in the
Affect the I

increases,

Where

and

In which

 

 

88

manufacturers decrease. This encourages the entry of new manufacturers in the South
and increases the fraction of goods produced in the South. As shown in Figure 3.4,
we have an increase in g and a decrease in 0. Similar results can be obtained in Lai.
In Liu’s model, as labor is used in both the actual manufacturing and the learning
activities, we can consider the increase in southern labor as the increase in southern
human capital. Thus, the positive growth effect of southern labor in his model is

qualitatively the same as the positive growth effect of the increase in southern human

capital in this section.

3.4.4 Increases in Southern Unskilled Labor (LS T)

Like in the southern human capital case, changes in southern unskilled labor do not
affect the northern equilibrium condition. Thus, the N N curve stays the same as Ls

increases. In Appendix, it is shown that total differentiation of (3.37) gives

 

511 < o if 0 > Q (3.39)
dLs s
where
0 :—: 1 — — > 1
7
and
_ 1
Q = 1 agﬂfp'l'g! > 1
l
in which
1 — a L

IE

 

(H5 — aLg)[B +(1— Mfg—Sam

89

As explained below, the larger the elasticity of substitution between goods in con-
sumption and the larger the elasticity of substitution between inputs in production,
the larger the value of 9 is. As 0 < a < 1 enters the elasticity of substitution between
goods in consumption, 6 E 1/(1 — a), larger values of 0:18 imply larger values of the
elasticity of substitution between goods. Also, as —1 S 7 S 0 enters the elasticity of
substitution between inputs in production, 5 E 1/ (1 — 7), smaller values of 719 imply
larger values of the elasticity of substitution between inputs. However, 9 is larger as
a is larger and 7 is smaller.

We can also say that, the more abundant southern human capital is, the closer
the value of Q is to 1. Note that, for a given value of g, Q becomes smaller, as I

becomes larger. However, manipulating I above and using (3.32), we can rewrite I as

l—a Hs-aLg
= [3
a ‘15s

I

 

The numerator in the last fraction becomes large, for a given g, as H 5 becomes large.
The denominator c155 measures the factor cost share of human capital in manufactur-
ing in the South. For a given g, relatively abundant southern human capital implies
a lower wage rate for human capital in the South. This makes the factor cost share of
human capital in manufacturing smaller. Thus, we can say that relatively abundant
southern human capital implies larger values of I and smaller values of Q. Therefore,

we can conclude that, for a given value of 0, sufficiently large amounts of southern

 

18Closer to 1.
19Closer to —1.

90

human capital ensure 0 > Q. Then, using (3.39), we can see that in this case the
SS curve will shift up. That is, for a sufﬁciently large enough stock of southern hu-
man capital, increases in southern unskilled labor increase the world growth rate and
the fraction of goods produced in the South. Increases in southern unskilled labor
lower the wage rate for southern unskilled labor. This tends to decrease the cost
of production for southern manufacturers. However, at the same time, increases in
southern unskilled labor increase the marginal product of human capital in manu-
facturing. This raises the demand and the wage rate for human capital. This tends
to offset the initial decrease in the manufacturing cost. Which of these two effects
dominates depends on the relative abundance of southern human capital. That is,
if human capital is relatively abundant in the South, the increase in the marginal
product of human capital would not be that high, and the decrease in manufacturing
cost through the decrease in the wage rate for unskilled labor dominates. Decreases
in southern manufacturing cost lower the price of the goods produced in the South.
This raises the demand, and thus the operating proﬁts, for southern goods. This
encourages entry of new manufacturers in the South and increases both the fraction
of goods produced in the South (0 l) and the rate of product developments (9 T).
This result is opposite to that in Lai’s model of imitation. In his model, increases
in southern unskilled labor also increase the marginal product of human capital in
southern manufacturing. Then, demand for human capital from southern manufac-
turers increases. This raises the wage rate for human capital, induces human capital
out of the southern research sector, and lowers the imitation rate. Decreases in the

imitation rate imply decreases in the hazard rate for northern manufacturers. This

tends to increase the innovation rate in the North. However, for a given number of
products, as less products are imitated by the South, more manufacturers are com-
peting for resources in the North. This lowers the per manufacturer operating profits.
This tends to decrease the innovation rate in the North. As the second effect dom-
inates the ﬁrst effect, in his model, increases in southern unskilled labor result in
decreases in 9. Using Lai’s word, the integration of labor abundant China into the
world trading economy unambiguously decreases the world growth rate. However,
ﬁndings in this model suggest that, if the pre-existing world trading economy is en-
dowed with enough human capital in the non-innovating South, the integration of
China into the southern economy would still have a positive growth effect.
Findings in this section can be summarized as the following

Theorem In an economy with North-South technology trade, an increase in the
supply of either northern or southern human capital increases the world growth rate.
An increase in the supply of southern unskilled labor also increases the world growth
rate, provided that South is endowed with relatively abundant human capital. However,

an increase in northern unskilled labor has a negative growth eﬂect.

3.5 Conclusion

Findings in this paper have many policy implications. The positive growth effect of
expanding human capital in each region has the implication related to the integration
of many eastern European economies into the western trading economy. During the

cold war, many communist countries tried to accumulate human capital to demon-

91

92

strate the superiority of their ideology. With the collapse of the old communists bloc,
a relatively rich stock of human capital is available to be integrated into the western
economy. Whether this human capital is distributed in the North or in the South,
we have. an unambiguously positive growth effect for the world economy. The sharp
increase in the fraction of high school graduates in the US. labor force, in 19803,
can be considered as an increase in unskilled labor force in the innovating North.
Then the negative growth effect of northern unskilled labor in this model can be an
explanation for the recent productivity slow down of the world economy. Integration
of relatively unskilled labor abundant China into the trading economy could have
positive growth effects for the world economy. However, for this result, we need to

have a relative abundance of human capital in pre-existing southern economy.

Ch

A.1

Bysu

ll We

wher
the l:

deter

Appendix A

Chapter 1

A.1 Consumers’ Expenditure Path

By substituting (1.3) into (1.2) and (1.2) in (1.1), we get

U :1.” e-ph-t){2n(s)ln[E(s)/2n(s)] —- {/OnM lnP(j,s)dj + An“) lnP(j',s)dj']}ds
(A.1)

If we maximize this subject to the intertemporal budget constraint

/°° e’lRl’l-thll . E(s)ds g Z(t) (A.2)
0

where Z (t) includes assets and tariff revenue distributed back to consumers along with
the present value of consumers’ wage incomes. These are either market or government

determined values and they are just like given from the consumers’ point of view.

93

Since the PI
also the git

problem is f

 

where u is ‘

Differentia

—p,e‘

By (livid

BF sett-

DER-1a:

be IOU)

94

Since the prices in the second bracket of the intertemporal utility function in (A.1) is

also the given values to consumers, the Hamiltonian Function for the maximization
problem is given as
H0 5 e“"(”‘)2n(s)ln[E(s)/2n(s)] — p - e'lRl‘)‘R("] - E(s)

where p is the usual costate variable. The maximum principle gives

2
e'pli-t) 138)) = p - e‘lm’l'm‘” (A3)

Differentiating (A.3) with respect to s, we get

. e-P("‘)M + e—p(s-t) N’(3)E(3) - N(3)EI(3) =

_p 13(8) 19(8)2 —RI(S)#e-[R(’)-R(i)l (AA)

By dividing (AA) by (A.3), we get

_ ,_ 21(3)
E(s)-R'() P+n(3) (A5)

 

By setting 3 = t and using 9 =_-' n’ / n, we have the expression in (1.4).

A.2 Final Goods Manufacturer

Demand for labor and intermediate goods from manufacturers of a ﬁnal good, j, can

be found by minimizing the cost

95

C(j) = wLy(j) + /: Pm(i):r(i)di + foj Pm(i"):r(i‘)di' (A.6)

subject to the production function given in (1.7). As the production function is

Cobb-Douglas type, we know that

wLyU) = (1 - B)C(J') (A-7)

and

f0” Pm(i):r:(z')di + [0’ Pm(i‘):r(i‘)dz" = 30(3) (A.8)

Combining (A.7) and (A8), we get

wLyv) = lﬁglfoj P..(i)x(z')dz' + /’ Pm(i')x(z")di'1 (A9)

Substituting (A9) into the cost function in (A.6), we get

C(j) = [131 f0 ’ Pm(i)x(i)di + [0’ Pm(i')x(i‘)di‘] (A.10)

As we combine two separate continuums i 6 [0, j] and i' E [0, j] into one continuum

v E [0, . . . ,j,j + 0,. . .,j +j], we can rewrite the cost function as

0(2) = g-I/ij P..(v)x(v)dv1 (A.11)

where

We can al

Rewritini

Let Ur)

{UHCIIOU

AS ‘6}

96

v=i ifv6[0,j]
v=i' ifvE (j,2j]

We can also rewrite the production function in (1.7) under one integral sign as

Y0) = Lifer-”IT,” were)”

Rewriting this, we get

[02” $(v)°'dv = [ﬁr/5 (A.12)

Let F(v) = f; a:(z)°’dz so that I‘(0) = 0 and I‘(2j) = [L—JquP/B. Then, we have
I‘ = z(v)dv

We can use this in place of the production function (A.12) as the constraint for the

minimization of cost function in (A.11). Noting 0 < ﬂ < 1 there, the Hamiltonian

function for this problem can be written as

Hp E Pm(v)x(v) — A2:(v)°'

As —3H/0I‘ = A = 0, we can treat A as a constant. We also get

97

6H
3x(v)

 

= Pm(v) — Acn:r:(v)‘""1 = 0

Rearranging this, we get

 

x(v)— — [P—(v—ja 1W 1 (A.13)
1 1: v a“
m = 1:30)) (A.14)

By substituting (A.13) in (A.12), we get

1 a a-l 2j a _ Y(j) a
(El/ [0 3(1)) d” - 1W} #3

Substituting from (A.14) rearranging the above, we get

 

1(1)): Pmlv )l/a-l Y8) 11/3

[fa-7pm (v)a/a-ldv]1/a . [Ly(j 1-3

If we rewrite this with separate integral signs for home and foreign intermediate goods

as in the original form, we get

' l/a—l -
x(i)_ Pm(z) Y(jl-B) 1w,

lfoPm i)°/°"1di + [0 Pm (i- )a/a’ldi‘ 11/0 ’[Lylj (A.15)

and

Substitutin

WU)

Substitutin

CU):

Differentiat

Cost Pricing

‘U

C

H
091 Q)

Inlefmediat

 

98

2(i') = Pm(i.)1/a-l . _YilLlllﬁ

lfoPm 00/0“le +1-de (1")0/0-1di-11/a [Ly(j)1-p (A.16)

Substituting (A.15) and (A.16) into (A9) and rearranging it, we get

     

wLy(j=) rot—W ﬁfﬂ/ Pz)( )°/°"di+/ P...( nah-1.15 "2‘ (j) (A.17)

Substituting (A.17) into (A.7), we get

° ”Y(J')

 

 

0(1) = (1- ﬂ)'("m3"’w"B[/oj Pm(i)°/°"dz' + j: Pm(i‘)°"°‘“di'

Differentiating this with respect to Y( j), we get the marginal cost function for the
ﬁnal goods manufacturing and perfect competition in this sector ensures the marginal

cost pricing for products. Thus,

P(j)= 3%:(1_’8)-(1-3)ﬁ'5w1-3[1)jPm(2°)or/a-ldi+'/0ij()_a/a_1dl.]°_-'g

(A.18)

Substituting (A.17) into (A.15) and (A.16) and using (A.18), we get the demand for

intermediate goods from the manufacturers of a ﬁnal good j as

99

 

 

:ri - Pm(i)l/a-l . ' '
( l ‘ [f3 Pm(i)°‘/°“di + f3 Pm(i')°/°“di'] (“OW“) (A '19)
and
it Ila—l
are"): P“ ) -ﬂP(j)Y(j) (A20)

U5 Pm(i)°/°'"di + If Pm(i")°/°“di'l

Substituting (A.19) and (A20) in (A9), we get the demand for labor from the

manufacturers of a ﬁnal good j as

. 1 - ﬂ . .
LY(J) = TP(J)Y(J) (A31)
As the ﬁnal good j is demanded both from the home and the foreign consumers,
we have Y(j) = C(j) + C‘(j) and P(j)Y(j) = E(j) + E'(j). As we normalize each
country’s expenditure level as E = E“ = l, we have expenditure for each ﬁnal product
as 1/2n as in (1.6). Noting this in (A.19), (A.20), and (A21), we get corresponding

expressions in (1.9), (1.10). and (1.8) in the main text.

Appendix B

Chapter 2

A representative consumer in this economy maximizes the discounted utility in (2.2)

subject to the dynamic budget constraint:

K=rK+w1L+wth+PAA+A1r—CL

and the differential equation in (2.1) that governs human capital accumulation. The

Hamiltonian function for this problem is

1—0

'HE e“" C

1—0

 

+ mm + w,L + wth + FAA + Arr — CL] + p2[AH(1— v)]
The equation of motion for the ﬁrst costate variable #1 is given by

_- -93-
Hl—aK-ﬂlr

100

101

Rearranging this, we get

_=_. (an

The ﬁrst order condition for the interior consumption path is

171
(9c

= 6-ptC—a — [11L = 0

Rearranging this, we get

e‘9‘c-a = p1L (B.2)

Differentiating (B.2) with respect to time, we get

-pt —0

—pe c — ae"“c"‘lc = [11L + mi. (3.3)

Dividing (3.3) by (3.2), we get

an. g
p OC_P1+L

Denoting % E n as the population growth rate and using (B.1), we can write the

above as in (2.3).

The equation of motion for the second costate variable, rig is given as

. (9%
‘112 = 5? = [1111)]{0 + [12A(1 — U) (3.4)

    
 

  
   

foalsbavti...r.¢. . .. .......o.:. t I,
.1213 u. 14s....» wt 5”...

1.1a magmas... .. _

  

102

The first order condition for an interior v is given as

67-1
'5'”- : plwhH -- pgAH ‘2 0

Rearranging, we get

#1 = 7112 (3.5)

Differentiating (B.5) with respect to time and dividing the result with (B.5) itself,

we get
ﬂ=&_ﬂ we
#1 #2 wk
Substituting (B.1) into (B.5), we get
-’-‘3 = r — 9-"- (13.7)
#2 wk

Substituting (B.5) into (8.4) and rearranging, we get

[12

Substituting (B.7) into above, we get the expression in (2.4).

Appendix C

 

 

 

 

 

Chapter 3
Let’s denote
h E (HN - aog)"”[ﬂ(H~ — aug)‘y + (1 - (3)163] (01)
1— L
Is aast - a.g)ia+ (1 — 3st _Sazgm (0.2)
and
fsArugr (cs
where
_ HS — 01.9
A = H” _ avg (0.4)
B - e 1 3 Ls ’1 c 5
C - 1 s L" 7 C 6
=IB+( - )(HN—aDg (i)
and
02a(1—a—1'—7)=1—3>1
a 7

104
Then, we can rewrite the northern and the southern equilibrium conditions, (3.29)

and (3.37), respectively, as

 

 

 

_ (1 — a)h
a - aﬂao(P + g) - (1 - C1)}! (07)
and
at. C.1-a

1' arﬂlp ‘1' g) = arﬂw + who/130, f (0'8)

C.1 Slopes of the N N and the SS Curves

Total differentiation of the northern equilibrium condition (C.7) gives

_ ah/ag aﬂap - (1 — a)0h/8g

d" ‘ ”l h ‘ cacao + g) — (1 - a)h]"g (0'9)

Partial differentiation of h in (C.1) with respect to g gives

h
$5 = we - aou - n(l — ﬁ)(H~ — ape-"Lt < 0

as 0 < a, ,8 < 1 and —1 S 7 _<_ 0. Noting this in (0.9), we have

do

— <0
dgN

as in the main text, and this means that the N N curve is downward sloping.

105

Total differentiation of the southern equilibrium condition (C.8) gives

 

2201-091
51/92., = _1__.g +0 .. .. d9
1 P+9 330‘“°f+(1+0)
(1 — a)%f~a‘°’f ﬁalwf

 

 

ital-ﬂ + (1 + a)“ _ (1+ allifamf + (I + ”llda

Rearranging terms, we have

gene!" — (1 + a)(1— one-u a _
(1+ anafal-‘rf +(1+ 0)] _

 

 

l ﬁUI-Gf af/ag al/ag
1.73:; + gal-«H (1 + a) f ‘ TM" (01°)

However, using (C.2)-(C.6), we have

 

% < 0 (0.11)
and
9%?! =(1_a)6—A//1—ag-+o[@§’i—9%931> 0 (0.12)
as
6.4469 > 0
and

33/09 _ BC/ag

B C>0

 

as long as If > an, ~15 > {if}, and —1 S 7 S 0. Noting (GM) and (C.12) in the

“D 0L

106

right hand side of (C.10), we can see that it is positive. Manipulating the term inside
the bracket of the numerator in the left hand side of (C.10), we can see that it is

positive when a > (1 — a)/a. Thus, assuming a > (1 — a)/a, If > :31, I: > 3:1

and—1S7S0,weget

do

dg >0

S

as in the main text, and this means that the SS curve is upward sloping.

C.2 Increases in Northern Human Capital (H N T)

Holding 9 and LN constant, total differentiation of both sides of (0.7) gives

“(1 - 0)300(P ‘1‘ g)_8h_ 3T1”

=laﬂao(P + 9) - (1 - a)hl’dHN

As partial differentiation of h in (C .1) with respect to H N gives

8h
6,, =ﬂ+(1-ﬁ)(1-)(H~—aog)"L

where the last inequality is assured by our assumptions 0 < S < l and —1 S 7 S 0,

we get

as in the main text.

107

Holding g, H5, LN, and L3 constant, total differentiation of (C8) gives

 

al 212(1 _. 000‘“de + 2R01"”33%LdHN 3110""de
d = CL CL N _ 1L
—aHN HN aLﬂ(p+g)l 1+0. (l+0’)2 ]

Noting 81/ aHN = 0 from (C.2), dividing the above equation by (C8), and rearranging

terms, we get

 

 

 

]d0' = dHN (C.13)

Using (C.3)-(C.6), we can also show that

Bf/aHN _ _ 6/1/6sz aB/BHN _ BC/BHN
——f — (1 a)——A + 0[-——B _C ] < 0 (C.14)
as
6A/6HN _ 1
A ‘ HN -aog < 0
BB/BHN _ 0
___—B _
and

 

60/0H~_ (l-ﬂ)~r(—-’1L)" ‘

_ _ HIV-009 LHN-aoa 0
C 5+(1-ﬂl7lﬂT-ﬁﬁli

Also the term in the bracket of the left hand side of (C.13) is positive under our

earlier assumption, 0 > (1 — a)/oz. Noting this along with (0.14) in (C.13), we have

d” < 0
ML

108

as in the main text.

C.3 Increases in Northern Unskilled Labor (L N T)

Holding g and H N constant, total differentiation of (Q7) gives

da’ _ (1— a)aSaD(p + 9) 8h

if; ‘ [er/sane + g) - (1 — an]? BLN (0°15)

From (C.1), we also get

8h

517,; = (1 - awn — anal-7L2,“ .<. o

Noting this in (0.15), we get

as in the main text.

Totally differentiating ((3.8), dividing the result by (C8), and noting al/BLN = 0

from (C.2), we get
1 1 - a _ af/BLN
1+ 0‘ 0’ ]d0' — ——f——dL~

I

 

(C.16)

As in the previous case, the term in the bracket of the left hand side is positive as

long as o > a/(l - 0). Also from (C.3)-(C.6),

Bf/BLN

f =(._.,w

A

68/6LN _ aC/aLN

+0[ B C

120

109

as
3A
an. ‘ 0
BB
6LN ‘ 0
and
80 LN 74 1
__ — _. ___... ___. <
aLN (1 BMIHN - any] HIV — avg - 0
Thus, from (C.16), we have
do
2 0
dLN s

as in the main text.

C.4 Increases in Southern Human Capital (H5 T)

Holding g, H N, LN, and L 5 constant, totally differentiating (C8), and dividing the

result by (C8), we get

1 1

_ af/BHS _ 31/6115 1
1 + a

[ f 1 l-a1ﬁ(P+9

 

 

 

;“wa=i

)1st (0.17)

As usual, terms in the left hand side bracket is positive. Using (C.3)-(C.6), we can

show that terms in the right hand side bracket becomes

110

(1—(3)(—Li— 7

IIs-dlw
n+(1— ﬂ)(—5—)} (0'18)

Hs-aw

 

1
milu - Q) - 0] - [(1 - Ql‘r - a]

where

= I
Q— l-arﬂUH-g) >1

 

With —1 < 7 _<_ 01, the absolute value of the ﬁrst bracket in (C.18) is larger than
that of the second bracket. As the term in the ﬁrst bracket is negative, we can say

that (C.18) is negative. Noting this in (C.17), we can conclude that

do
EELS < 0

as in the main text.

C.5 Increases in Southern Unskilled Labor (L 5 T)

Totally differentiating (C8) and dividing the result by (C8), we get

 

         

 

 

1_0 3L Bl 3L 1
[ (f—— 5— / 5 ]dLs (c.19)
1+0 f I I-arﬂ(p+g)
1When 7 = —l, absolute values of bracketed terms are equal. However, the fraction terms

attached to the second bracket still guarantees that terms in (C.18) is negative.

111

As usual, the term in the left hand side bracket is positive. Using (C.3)-(C.6), we can

show that terms in the right hand side bracket become

af/aLs _ 61/6Ls I (1- P)(—LL)” 1

Hs-atg

f I I-a.(3(p+g)]=(9‘Ql73+(1—ﬂ)(—Ls—)~E

Hs-ﬂw

l

 

 

(0.20)

Thus, the sign of (C.20) depends on the relative size of 9 and Q. That is, if 0 > Q,

we can conclude that (C.20) is negative and, noting this in (C.19), we can say that
d
—g- < 0 if 0 > Q
dLs s

as in the main text.

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